The substitution u = 3x transforms the integral 31 de into: ∫(3/31)du
This is because the substitution u = 3x implies that du/dx = 3, which means dx = (1/3)du. Substituting this expression for dx in the original integral and using the fact that e is a constant, we have:
∫e^(3x) dx = ∫e^(u) (1/3)du = (1/3)∫e^(u)du = (1/3)e^u + C
where C is the constant of integration. So the final answer in terms of x is:
(1/3)e^(3x) + C
which is equivalent to the original integral. This is an example of how the technique of substitution can be used to simplify an integral and make it easier to solve. It is also a common step in many integral transforms.
When performing an integral with substitution, you transform the original integral into a new one with a different variable. In your case, the substitution is given as u = 3x.
To apply substitution, first find the derivative of the substitution equation with respect to x, which is du/dx = 3. Then, solve for dx: dx = du/3.
Now, substitute u = 3x into the original integral and replace dx with du/3. The transformed integral will have the new variable u and a constant factor 1/3.
Without knowing the specific function you're trying to integrate (it seems like "31 de" might be a typo), I cannot provide the exact transformed integral.
However, I hope this explanation of substitution and transforming integrals is helpful. Please feel free to provide more information or clarify your question if you need further assistance!
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Find the area of the regular polygon with the given apothem a and side length s.
pentagon, a = 10. 4 cm, s = 15. 1 cm
The area of the regular pentagon is approximately 392.2 square centimeters.
The area of a regular polygon can be calculated using the formula:
A = (1/2) * apothem * perimeter
where perimeter = number of sides * side length.
For a pentagon with side length s = 15.1 cm, the perimeter is:
perimeter = 5 * 15.1 = 75.5 cm
The apothem is a = 10.4 cm.
Using the formula, we get:
A = (1/2) * 10.4 * 75.5
A = 392.2 cm^2
Therefore, the area of the regular pentagon is approximately 392.2 square centimeters.
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A searchlight is shaped like a paraboloid of revolution. if the light source is located 1 feet from the base along the axis of symmetry and the opening is 6 feet across, how deep should the searchlight be?
The searchlight should be 1/3 feet deep at the edge of the opening. Since the paraboloid is a continuous surface, the depth will increase gradually from the edge of the opening to the vertex at (0,0,1).
Determine the depth of the searchlight shaped like a paraboloid of revolution, we need to use the equation for the standard form of a paraboloid of revolution:
z = (x^2 + y^2) / (4f)
where z is the depth, x and y are the horizontal and vertical coordinates, and f is the focal length of the paraboloid.
We know that the light source is located 1 feet from the base along the axis of symmetry, which means that the vertex of the paraboloid is at (0,0,1).
We also know that the opening is 6 feet across, which means that the horizontal distance from one side of the opening to the other is 3 feet.
Using this information, we can find the value of f:
f = (d/2)^2 / 2r
where d is the diameter of the opening (6 feet), and r is the radius of curvature at the vertex (1 foot).
f = (6/2)^2 / 2(1) = 4.5 feet
Now we can plug in the values for x, y, and f to solve for z:
z = (x^2 + y^2) / (4f)
z = (x^2 + y^2) / (4(4.5))
z = (x^2 + y^2) / 18
Since the opening is 6 feet across, we know that the maximum value of x is 3 feet. Therefore, we can use the maximum value of y (also 3 feet) to find the depth at the edge of the opening:
z = (3^2 + 3^2) / 18
z = 6/18
z = 1/3 feet
So the searchlight should be 1/3 feet deep at the edge of the opening. However, since the paraboloid is a continuous surface, the depth will increase gradually from the edge of ×the opening to the vertex at (0,0,1).
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How would you do a point circle problem like this without arctan?
To do this, we can use the Pythagorean theorem and trigonometric ratios instead.
1. Determine the coordinates of the given point, let's call it P(x, y), and the center of the circle, let's call it O(h, k). Also, note the radius, r.
2. Calculate the distance between point P and the center O using the Pythagorean theorem: d^2 = (x-h)^2 + (y-k)^2, where d is the distance.
3. Set d equal to the radius of the circle: r^2 = (x-h)^2 + (y-k)^2.
4. Now, let's find the angle θ between the x-axis and the line OP without using arctan. To do this, we'll use the sine and cosine ratios:
sin(θ) = (y-k) / r and cos(θ) = (x-h) / r
5. To eliminate the need for arctan, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1. Substitute the sine and cosine ratios we found earlier:
((y-k) / r)^2 + ((x-h) / r)^2 = 1
6. Simplify the equation by multiplying both sides by r^2:
(y-k)^2 + (x-h)^2 = r^2
You'll notice that this equation is the same as the one we found in step 3, confirming that the point P lies on the circle. You've now solved the point circle problem without using arctan, by employing the Pythagorean theorem and trigonometric ratios instead.
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In ΔMNO, the measure of ∠O=90°, OM = 1. 7 feet, and NO = 6. 7 feet. Find the measure of ∠N to the nearest degree
The measure of ∠N to the nearest degree is 82°.
Calculate the nearest degree?To find the measure of ∠N, we can use the Pythagorean theorem and trigonometric functions.
First, we can use the Pythagorean theorem to find the length of MN:
MN² = NO² - OM²
MN² = (6.7 feet)² - (1.7 feet)²
MN² = 44.56 feet²
MN = 6.67 feet
Now, we can use the sine function to find the measure of ∠N:
sin(N) = MN/NO
sin(N) = 6.67 feet / 6.7 feet
sin(N) ≈ 0.994
N ≈ sin⁻¹(0.994)
N ≈ 82.4°
The measure of ∠N to the nearest degree is 82°.
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Choose the system for the graph.
The system of inequalities in the graph is the one in option A.
y ≥ (-2/5)x - 2/5
y ≥ (3/2)*x - 1
Which is the system of inequalities in the graph?Here we can see the graph of a system of inequalities, on the graph we can see two lines.
The first one is a line with a positive slope, it has an y-intercept of -1, the shaded region is above that line, and it is a solid line, so one of the inequalities is:
y ≥ a*x - 1
Where a is positive.
The second line has a negative slope, and we can see that the shaded region is also above the line, so this second inequality is like:
y ≥ line with negative slope.
It is easy to identify the correct option because there is only one with these properties, which is the first option:
y ≥ (-2/5)x - 2/5
y ≥ (3/2)*x - 1
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8.4 Could be the Hypotenuse, Could be a
Leg
PLSSSSSSSSS HELPPPPPPPPPPPP
Answer:
1. x = 5
2. x = [tex]\sqrt 7 \approx[/tex] 2.65
Step-by-step explanation:
To solve these problems, we can use the diagram below, as well as Pythagoras's Theorem, which states:
[tex]\boxed{\mathrm{a^2 = b^2 + c^2}}[/tex],
where a is the hypotenuse (longest side) of a right-angled triangle, and b and c are the legs of the triangle.
1. x is the hypotenuse:
From the diagram below, we can see that, if x is the hypotenuse of the triangle, then 3 and four are the legs of the triangle. Therefore, using the above equation:
[tex]x^2 = 3^2 + 4^2[/tex]
⇒ [tex]x = \sqrt{3^2 + 4^2}[/tex] [Taking the square root of both sides of the equation]
⇒ [tex]x = \sqrt{25}[/tex]
⇒ [tex]x = \bf 5[/tex]
2. x is one of the legs:
From the diagram below, we can see that when x is one of the legs, the hypotenuse is 4, and the other leg is 3.
The hypotenuse isn't 3 because the hypotenuse is the longest side in a right-angled triangle, and 4 is longer than 3.
Therefore,
[tex]4^2 = x^2 + 3^2[/tex]
⇒ [tex]x^2 = 4^2 - 3^2[/tex] [Subtracting 3² from both sides]
⇒ [tex]x = \sqrt{4^2 - 3^2}[/tex] [Taking the square root of both sides of the equation]
⇒ [tex]x = \sqrt{7}[/tex]
[tex]\approx \bf 2.65[/tex]
A 2 column by 10 row table. column 1 is titled how energy is used with the following entries: computers, cooking, electronic, lighting, refrigeration, space cooling, space heating, water heating, wet cleaning, other. column 2 is titled energy used in percent with the following entries: 2, 4, 5, 6, 4, 9, 45, 18, 3, 7. a carbon footprint is a measure of the amount of carbon human activities, like using energy, release into the atmosphere. which activity would help decrease the greatest carbon-releasing activity in us homes? limiting time in hot showers wearing layers of clothing turning off lights when leaving a room unplugging electronics when not in use
Limiting space heating would help decrease the greatest carbon-releasing activity in US homes.
From the table, we can see that space heating accounts for the largest percentage of energy use in US homes at 45%. Therefore, by limiting space heating, we can significantly reduce the amount of carbon released into the atmosphere, thus decreasing our carbon footprint.
Other activities like limiting time in hot showers, wearing layers of clothing, and turning off lights and electronics when not in use can also help reduce our carbon footprint, but they are not as effective as limiting space heating.
Additionally, we can consider using energy-efficient heating systems, improving insulation, and reducing air leaks to further reduce our energy use and carbon footprint.
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A laundry basket contains 14 socks, of which 4 are blue. What is the probability that a randomly selected sock will be blue? Write your answer as a fraction or whole number.
Answer:
the probability of selecting a blue sock from the laundry basket is 2/7 or approximately 0.2857.
Step-by-step explanation:
The probability of selecting a blue sock can be found by dividing the number of blue socks by the total number of socks in the basket:
Probability of selecting a blue sock = Number of blue socks / Total number of socks
Probability of selecting a blue sock = 4 / 14
Simplifying the fraction by dividing both the numerator and denominator by 2 gives:
Probability of selecting a blue sock = 2 / 7
What is the money multiplier when the reserve requirement is:
(Instructions: Enter your responses rounded to three decimal places.)
(a) 0.040?
(b) 0.125?
(c) 0.400?
(d) 0.200?
The money multipliers are:
(a) 25.000
(b) 8.000
(c) 2.500
(d) 5.000
The money multiplier represents the amount of money the banking system can create through the lending process for every dollar of reserves held by the central bank. It is inversely related to the reserve requirement, which is the percentage of deposits that banks are required to hold as reserves.
When the reserve requirement is low, su
The money multiplier is given by the formula:
Money multiplier = 1 / Reserve requirement
(a) When the reserve requirement is 0.040, the money multiplier is:
Money multiplier = 1 / 0.040 = 25.000
(b) When the reserve requirement is 0.125, the money multiplier is:
Money multiplier = 1 / 0.125 = 8.000
(c) When the reserve requirement is 0.400, the money multiplier is:
Money multiplier = 1 / 0.400 = 2.500
(d) When the reserve requirement is 0.200, the money multiplier is:
Money multiplier = 1 / 0.200 = 5.000
Therefore, the money multipliers are:
(a) 25.000
(b) 8.000
(c) 2.500
(d) 5.000
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A ferry is used to transport guests from the dock to two hotels across a large lake. The hotels are located 550 m apart. The first hotel is at a 49 angle between the dock and the second hotel. The second hotel is at a 56 angle between the dock and the first hotel. How far is each hotel from the dock?
The first hotel is approximately 0.6246 meters away from the dock, and the second hotel is approximately 0.4931 meters away from the dock.
To solve this problem, we can use trigonometry and create a system of equations based on the given angles and distances. Let's assume the distance between the dock and the first hotel is x meters and the distance between the dock and the second hotel is y meters.
Using the law of sines, we can relate the angles and distances:
For the first hotel:
sin(49°) = y / x ...(Equation 1)
For the second hotel:
sin(56°) = x / y ...(Equation 2)
We can rearrange Equation 1 to solve for y:
y = x * sin(49°)
Substituting this value of y into Equation 2:
sin(56°) = x / (x * sin(49°))
sin(56°) = 1 / sin(49°)
Now, we can solve for x by isolating it:
x = sin(56°) * sin(49°)
Plugging in the values and evaluating the equation:
x = 0.8290 * 0.7539
x ≈ 0.6246
Therefore, the distance between the dock and the first hotel (x) is approximately 0.6246 meters.
To find the distance between the dock and the second hotel (y), we can substitute this value back into Equation 1:
y = 0.6246 * sin(49°)
y ≈ 0.4931
Hence, the distance between the dock and the second hotel (y) is approximately 0.4931 meters.
In summary, the first hotel is approximately 0.6246 meters away from the dock, and the second hotel is approximately 0.4931 meters away from the dock.
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Find the curve in the xy plane that passes through the point (9,8) and whose slope at each point is 6 sqrt x
The equation of curve passing through the point (9,8) with slope [tex]6\sqrt{x}[/tex] is [tex]y = 4\times x^{(3/2)} -100[/tex] .
Slope of curve is given by [tex]6\sqrt{x}[/tex] and curve is passing through point (9,8) .
A curve can be represented in a graph using the standard form of equations. Equation will represent the slope of curve and point through which the curve is passing.
Equation of curve :
Differentiate the slope equation,
dy/dx = [tex]6\sqrt{x}[/tex]
dy = [tex]6\sqrt{x}[/tex] dx
Integrating both sides,
Integration rule : [tex]\int\ {x^n} \, dx = x^{n+1}/n+1 + c[/tex]
[tex]y = 6 \times (x^{3/2})/(3/2) +c[/tex]
[tex]y = 4 x^{(3/2)} + c[/tex]
Now substitute (9,8) in the equation of y,
[tex]8 = 4\times (9)^{3/2} + c[/tex]
[tex]c = -100[/tex]
Substitute the value to get the equation of curve,
The equation of curve is [tex]y = 4\times x^{(3/2)} -100[/tex] .
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You are fishing for surfperch and rockfish, which are species of bottomfish. Gaming laws allow you to catch no more than 15 surfperch per day, no more than 10 rockfish per day, and no more than 20 total bottomfish per day.
Write a system of linear inequalities that represents the situation. Let a represent the number of surfperch and y represent the number of rockfish
Surfperch inequality:
Rockfish inequality:
Bottomfish inequality:â
The system of linear inequalities representing the fishing restrictions is a ≤ 15, y ≤ 10, a + y ≤ 20.
How to find the Surfperch inequality, Rockfish inequality, and Bottomfish inequality?Let's define the variables as follows:
a = Number of surfperch caught per day.
y = Number of rockfish caught per day.
Based on the given information, we can write the following system of linear inequalities:
Surfperch inequality: a ≤ 15
This inequality represents the restriction that you cannot catch more than 15 surfperch per day.
Rockfish inequality: y ≤ 10
This inequality represents the restriction that you cannot catch more than 10 rockfish per day.
Bottomfish inequality: a + y ≤ 20
This inequality represents the overall restriction that the total number of bottom fish (which includes both surfperch and rockfish) cannot exceed 20 per day.
Therefore, the system of linear inequalities is:
a ≤ 15
y ≤ 10
a + y ≤ 20
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The regular polygon has the following measures.
a = 2√3 yd
s = 4 yd
Segment a is drawn from the center of the polygon
perpendicular to one of its sides.
What is the vocabulary term for segment a?
What is the area of the polygon?
Round to the nearest tenth and include correct units.
Show all your work.
The vocabulary for the segment a is the apothem
The area of the polygon is about 41.6 yd²
What is the area of a regular figure?The area of a regular figure is the extent of the planer space the figure occupies.
The length of each side of the regular polygon, s = 4 yd
The length of the segment a = 2·√3
The vocabulary term for the segment a drawn from from the center of the polygon and perpendicular to one of its sides is the apothem
Therefore, the vocabulary term for segment a is the apothem
The polygon is a hexagon.
The area of a hexagon is; A = ((3·√3)/2) × s²
Therefore, the area of the polygon is; A = ((3·√3)/2) × (4)² = 24·√3 ≈ 41.6
The area of the polygon is about 41.6 yd²
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For each of the following lengths, estimate the perimeter of an isosceles right triangle whose short sides have that length
A. Length of shirt sides is 0. 75
The perimeter of an isosceles right triangle whose short sides have that length 0.75 units is 2.56 units.
Given that, an isosceles right triangle whose short sides have that length 0.75 units.
Let the longest side be x.
Here, x²=0.75²+0.75²
x²=1.125
x=√1.125
x=1.06 units
Now, the perimeter = 0.75+0.75+1.06
= 2.56 units
Therefore, the perimeter of an isosceles right triangle whose short sides have that length 0.75 units is 2.56 units.
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Find the next term in each sequence.
Question 1:
0, 1, 3, 7, ? .
Question 2:
35, 33, 29, 21, ?.
Please Include an Explanation of how to solve problems like this!
Thanks a ton!
Full Boat Manufacturing has projected sales of $115. 5 million next year. Costs are expected to be $67. 4 million and net investment is expected to be $12. 3 million. Each of these values is expected to grow at 9 percent the following year, with the growth rate declining by 1 percent per year until the growth rate reaches 5 percent, where it is expected to remain indefinitely. There are 4. 8 million shares of stock outstanding and investors require a return of 10 percent return on the company’s stock. The corporate tax rate is 21 percent
The given question is incomplete, the complete question is given
" Full Boat Manufacturing has projected sales of $115 million next year. Costs are expected to be $67 million and net investment is expected to be $12 million. Each of these values is expected to grow at 14 percent the following year, with the growth rate declining by 2 percent per year until the growth rate reaches 6 percent, where it is expected to remain indefinitely. There are 5.5 million shares of stock outstanding and investors require a return of 13 percent on the company’s stock. The corporate tax rate is 21 percent.
What is your estimate of the current stock price?
The estimate of the current stock price is $13.11 per share.
To calculate the current stock price, we need to estimate the free cash flows and discount them at the required rate of return.
First, we determine the firm's free cash flow (FCFF) for the following year.
FCFF = Sales - Costs - Net Investment*(1-t)
= $115 million - $67 million - $12 million*(1-0.21)
= $31.02 million
Now, we will calculate the expected growth rate in FCFF
g = (FCFF Year 2 / FCFF Year 1) - 1
FCFF Year 2 = FCFF Year 1 × (1 + g)
g = (FCFF Year 2 / FCFF Year 1) - 1
= (FCFF Year 1 × (1 + 0.14) × (1 - 0.02) / FCFF Year 1) - 1
= 0.12
We can now use the Gordon growth model to estimate the current stock price
Current stock price = FCFF Year 1 × (1 + g) / (r - g)
r = required rate of return.
Current stock price = $31.02 million × (1 + 0.12) / (0.13 - 0.12)
= $72.13 million
Finally, we divide the current stock price by the number of shares outstanding to get an estimate of the current stock price per share:
Current stock price per share = $72.13 million / 5.5 million
= $13.11 per share
Therefore, the estimate of the current stock price is $13.11 per share.
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This question please
Box of candy contains 0. 6 of a pound of caramels 3. 6 pounds of coconut What percent the contents of the box, by weight consists of caramels?
The contents of the box, by weight, consists of 14.29 percent caramels.
We need to find the percentage of caramels in the box, given the weights of caramels and coconut candies.
Step 1: Determine the total weight of the candies in the box.
The box contains 0.6 pounds of caramels and 3.6 pounds of coconut candies. Add these two weights together:
Total weight = 0.6 (caramels) + 3.6 (coconut)
Total weight = 4.2 pounds
Step 2: Calculate the percentage of caramels in the box.
To find the percentage, divide the weight of caramels by the total weight of the box and then multiply by 100:
Percentage of caramels = (Weight of caramels / Total weight) x 100
Percentage of caramels = (0.6 / 4.2) x 100
Step 3: Solve the equation.
Percentage of caramels = (0.6 / 4.2) x 100 ≈ 14.29%
So, approximately 14.29% of the contents of the box, by weight, consists of caramels.
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The rule is x→y=−2x. Can you mark all the points that fit the rule on a coordinate plane?
Yes, the points that fit the rule x→y=−2x can be plotted or marked on a coordinate plane, and will form a straight line passing through the origin with a slope of -2.
The rule x→y=−2x means that for every value of x, the corresponding value of y is equal to -2 times x. To plot the points that fit this rule, we can choose some values of x, substitute them into the equation, and then plot the resulting points on the coordinate plane.
For example, if we choose x=1, then y=−2x=−2(1)=−2. So the point that fits the rule is (1, −2). Similarly, if we choose x=2, then y=−2x=−2(2)=−4. So the point that fits the rule is (2, −4).
We can continue this process for other values of x, and plot all the resulting points on the coordinate plane. The resulting graph will be a straight line that passes through the origin, with a slope of -2. This means that all the points that fit the rule will lie on this line.
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How would the variation that exists between the distance that a man walks in one hour and the rate at which the man is walking be described? f inversely g directly h both inversely and directly j irregular
The relationship between these two variables can be described as inversely proportional.
The variation between the distance that a man walks in one hour and the rate at which the man is walking can be described as inversely proportional.
This means that as the rate at which the man is walking increases, the distance that he walks in one hour decreases. Similarly, as the rate at which he is walking decreases, the distance that he walks in one hour increases.
Therefore, the relationship between these two variables can be described as inversely proportional.
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After an antibiotic is taken, the concentration of the antibiotic in the bloodstream is modeled by the function C(t) = 4te-397, where t is measured in hours and C is measured in ag Use the closed interval methods to mg detremine the maximum concentration of the antibiotic between hours 1 and 7. Write a setence stating your result, round answer to two decimal places, and include units.
To find the maximum concentration of an antibiotic between hours 1 and 7, first find the critical points of the function C(t), then evaluate C(t) at the critical points and endpoints to choose the highest value.
To determine the maximum concentration of the antibiotic between hours 1 and 7, follow these steps:
1. Find the critical points of the function C(t) = 4te^(-397). To do this, find the first derivative of the function, C'(t), and set it equal to 0.
2. Check the value of C(t) at the critical points and the endpoints of the interval, t=1 and t=7.
3. Choose the highest value of C(t) among the critical points and the endpoints.
1: Find the first derivative, C'(t).
C(t) = 4te^(-397)
C'(t) = 4e^(-397)(1-397t)
2: Set the first derivative equal to 0 and solve for t.
4e^(-397)(1-397t) = 0
1 - 397t = 0
t = 1/397
3: Evaluate C(t) at the critical point t = 1/397 and the interval endpoints t = 1 and t = 7.
C(1/397) = 4(1/397)e^(-397(1/397)) ≈ 0.01 ag/mg
C(1) = 4(1)e^(-397(1)) ≈ 0.00 ag/mg
C(7) = 4(7)e^(-397(7)) ≈ 0.00 ag/mg
The maximum concentration of the antibiotic occurs at t = 1/397 hours, with a concentration of approximately 0.01 ag/mg. What is Titration: Titration is a technique by which we know the concentration of unknown solution using titration of this solution with solution whose concentration is known.
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A window in the shape of a parallelogram has a base of 36 inches, and a height of 45 inches. What is the area?
The area of a parallelogram is given by the formula:
$\sf\implies{\boxed{A = bh}}$
where $b$ is the length of the base and $h$ is the height.
In this case, the base is 36 inches and the height is 45 inches, so the area of the parallelogram is:
$\sf\implies\:A = bh = 36 \cdot 45 = 1620$
Therefore, the area of the parallelogram-shaped window is 1620 square inches.
[tex]\begin{align}\colorbox{black}{\textcolor{white}{\underline{\underline{\sf{Please\: mark\: as\: brillinest !}}}}}\end{align}[/tex]
[tex]\textcolor{lime}{\small\textit{If you have any further questions, feel free to ask!}}[/tex]
[tex]{\bigstar{\underline{\boxed{\sf{\color{red}{Sumit\:Roy}}}}}}\\[/tex]
Answer:
1620 inches ^2
Step-by-step explanation:
The area of a parallelogram is simply put as:
A=bh
where b is base and h is height
Given b is 36 and 45 is our h, we can now solve for the area.
A=(36)(45)
A=1620
Friendly reminder:
When multiplying two of the same units, remember to square them to have the correct labeling, so in conclusion, our answer is:
1620 inch.^2
Use this information for Ms. Yamagata is going to tile the floor of her rectangular bathroom that is 9 feet long and 73 feet wide. The cost per 6-inch tile is $0. 50. The cost per 18-inch tile is $2. 75. 4. If Ms. Yamagata uses 6-inch tiles, what are the least number of tiles that she needs to buy to cover the floor? оâ
Ms. Yamagata needs to buy at least 2628 6-inch tiles to cover the floor of her rectangular bathroom.
To determine the least number of 6-inch tiles Ms. Yamagata needs to buy to cover her 9 feet long and 73 feet wide bathroom floor, follow these steps:
1. Convert the dimensions of the bathroom to inches, as the tiles are measured in inches:
9 feet * 12 inches/foot = 108 inches long
73 feet * 12 inches/foot = 876 inches wide
2. Determine the total area of the bathroom in square inches:
Area = length * width = 108 inches * 876 inches = 94,608 square inches
3. Calculate the area of a single 6-inch tile:
Area = length * width = 6 inches * 6 inches = 36 square inches
4. Divide the total area of the bathroom by the area of a single tile to find the least number of tiles needed:
Number of tiles = total area / tile area = 94,608 square inches / 36 square inches ≈ 2,628.56
Since Ms. Yamagata cannot buy a fraction of a tile, she needs to buy at least 2,629 6-inch tiles to cover her bathroom floor.
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Victoria has $200 of her birthday gift money saved at home, and the amount is modeled by the function h(x) = 200. She reads about a bank that has savings accounts that accrue interest according to the function s(x) = (1. 05)x−1. Explain how Victoria can combine the two functions to model the total amount of money she will have in her bank account as interest accrues after she deposits her $200. Justify your reasoning.
WILL GIVE BRANLIEST
The reasoning behind this is that the initial amount (h(x)) is multiplied by the interest growth factor (s(x)) to calculate the total amount with interest over time.
To model the total amount of money Victoria will have in her bank account as interest accrues after depositing her $200, you can combine the two functions h(x) and s(x). The given functions are h(x) = 200 and s(x) = (1.05)^x−1.
First, note that s(x) represents the growth factor of the interest, which is 5% (1.05) compounded annually. To find the total amount after x years, you need to multiply the initial amount by the growth factor raised to the power of x.
So, the combined function T(x) can be written as T(x) = h(x) * s(x).
Substitute the given functions into the combined function:
T(x) = (200) * ((1.05)^x−1)
This function, T(x), models the total amount of money Victoria will have in her bank account after x years with interest accrued on her $200 deposit. The reasoning behind this is that the initial amount (h(x)) is multiplied by the interest growth factor (s(x)) to calculate the total amount with interest over time.
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You want to build a fence for a rectangular dog run. You want the run to be at least 10 ft wide. The run can be at most 50 ft long. You have 126 ft of fencing. Write a system of inequalities that describes the situation.
The system of inequalities that models the situation is given as follows:
w ≥ 10.0 < l ≤ 50.2w + 2l ≤ 126.What is the perimeter of a polygon?The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.
The perimeter of a rectangle of width w and length l is given as follows:
P = 2w + 2l.
You want the run to be at least 10 ft wide, hence:
w ≥ 10.
The run can be at most 50 ft long, hence:
0 < l ≤ 50.
(length has to be greater than zero).
You have 126 ft of fencing, hence the perimeter is represented as follows:
2w + 2l ≤ 126.
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A recipe requires only blueberries and strawberries. This list shows the amounts required for 1/4 of the whole recipe:
1/2 cup blueberries
2/5 cup of strawberries
What is the number of cups of blueberries and the number of cups of strawberries required for the whole recipe?
a) 1/8 cup of blueberries and 1/10 cup of strawberries
b) 1/8 cup of blueberries and 1 3/5 cups of strawberries
c) 2 cups of blueberries and 1/10 cup of strawberries
d) 2 cups of blueberries and 1 3/5 cups of strawberries
A
Either divide each by one fourth or multiply each by 0.25. Then turn the answer to a fraction.
Luca places $2,000 in an account that earns 2. 5% nominal yearly interest, compounded quarterly. Which of
the following is closest to the amount that the account is worth after 15 years if no additional deposits nor
withdrawals are made?
(1) $2,751. 08
(3) $2,853. 75
(2) $2,812. 19
(4) $2,906. 59
The closest answer to the amount that the account is worth after 15 years is $2,812.19, Therefore Option 3 is correct
The formulation for calculating the future value (FV) of an investment with compound interest is:
[tex]FV = P * (1 + r/n)^{(n*t)}[/tex]
Wherein P is the primary (the initial amount invested), r is the once a year interest rate, n is the variety of times the interest is compounded per year, and t is the term in years.
In this case, P = $2,000, r = 2.5% = 0.0.5, n = 4 (for the reason that interest is compounded quarterly), and t = 15. Plugging those values into the formulation, we get:
[tex]FV = $2,000 * (1 + 0.1/2/4)^{(4*15)}[/tex]
FV ≈ $2,812.19
Therefore, the closest answer to the amount that the account is worth after 15 years is $2,812.19.
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if you increase your reading speed so that each page takes you 30 seconds less than it did before, and you begin reading 20 minutes per day, how many 200 page books can you now read in a year
We can read 91 books in a year if you increase your reading speed so that each page takes you 30 seconds less than it did before.
How is the number of books calculated?Now If each page now takes 30 seconds less to read than before,
then you will save 30*200 = 6000 seconds (or 100 minutes) on each book.
So, the time it will take you to read a 200-page book will be
20 minutes - 100 minutes = -80 minutes,
which means you will finish a 200-page book in 80 minutes (or 1 hour and 20 minutes).
In a year, there are 365 days. If you read for 20 minutes per day, then you will read for a total of
365 * 20 = 7300 minutes (or 121.67 hours) in a year.
Since you can finish a 200-page book in 80 minutes, you can read 7300/80 = 91.25 books in a year.
However, since you cannot read a fraction of a book, the maximum number of 200-page books you can read in a year is 91 books.
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Complete question is :
If you increase your reading speed so that each page takes you 30 seconds less than it did before,
and you begin reading 20 minutes per day,
How many 200 page books can you now read in a year?
The function f(x) = 2x^3 +30x^2 – 54x +5 has one local minimum and one local maximum. This function has a local minimum at x = with value
and a local maximum at x =
with value
The local maximum is at x = -9 with a value of 2575, and the local minimum is at x = 1 with a value of -17.
To find the local minimum and maximum of the given function, we need to find its critical points, where the derivative is zero or undefined.
[tex]f(x) = 2x^3 + 30x^2 - 54x + 5[/tex]
[tex]f'(x) = 6x^2 + 60x - 54[/tex]
Setting f'(x) = 0, we get:
[tex]6x^2 + 60x - 54 = 0[/tex]
[tex]x^2 + 10x - 9 = 0[/tex]
(x + 9)(x - 1) = 0
x = -9 or x = 1
Now, we need to determine if these critical points correspond to local minimum or maximum.
To do so, we can use the second derivative test. We calculate the second derivative of f(x):
f''(x) = 12x + 60
At x = -9:
f''(-9) = 12(-9) + 60 = -48 < 0
This means that f(x) has a local maximum at x = -9.
At x = 1:
f''(1) = 12(1) + 60 = 72 > 0
This means that f(x) has a local minimum at x = 1.
To find the values of the local minimum and maximum, we plug in the corresponding x-values into the original function:
[tex]f(-9) = 2(-9)^3 + 30(-9)^2 - 54(-9) + 5 = 2575[/tex]
[tex]f(1) = 2(1)^3 + 30(1)^2 - 54(1) + 5 = -17[/tex]
Therefore, the local maximum is at x = -9 with a value of 2575, and the local minimum is at x = 1 with a value of -17.
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A gift bag is shaped like a rectangular prism and has a volume of 1152 cubic inches.
The volume of the gift bag is given as 1152 cubic inches.
Since it is shaped like a rectangular prism, we can write the formula for its volume as V = l × w × h, where l, w, and h are the length, width, and height of the rectangular prism, respectively.
To determine the dimensions of the gift bag, we need more information such as the ratio of its length, width, and height or any one of its dimensions. If we assume one of the dimensions, say, the length is L inches, then we can write the volume as V = L × w × h. Solving for w × h, we get w × h = V/L = 1152/L.
We can then use this equation along with the fact that the gift bag is a rectangular prism to find the other dimensions. For example, if the width is W inches, then we have h = 1152/(L × W) and the volume can be expressed as V = L × W × 1152/(L × W) = 1152.
Similarly, if the height is H inches, then we have w = 1152/(L × H) and the volume can be expressed as V = L × 1152/(L × H) × H = 1152.
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What are the coordinates of the point on the directed line segment from (-3,-5) to (9,−8) that partitions the segment into a ratio of 2 to 1?
The coordinates of the point are (7,0).
How to solve for the coordinatesdistance from (-3,-5) to (x,y) = 2 * distance from (x,y) to (9,-8)
Using the distance formula, we can write this equation as:
√[(x - (-3))^2 + (y - (-5))^2] = 2 * √[(9 - x)^2 + (-8 - y)^2]
Simplifying this equation, we get:
[tex](x + 3)^2 + (y + 5)^2 = 4[(9 - x)^2 + (-8 - y)^2][/tex]
Expanding and simplifying further, we get:
[tex]17x + 16y = 119[/tex]
So the coordinates of the point on the directed line segment from (-3,-5) to (9,-8) that partitions the segment into a ratio of 2 to 1 are:
x = (119 - 16y)/17
y = any value (since we can choose any value of y and then calculate x using the equation above)
For example, if we choose y = 0, then we get:
x = (119 - 16(0))/17 = 7
So the coordinates of the point are (7,0).
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