It would take approximately 95 minutes for the wick to have 74 mm remaining.
1) Finding the slope (mm per minute):
The wick was initially 150 mm in length and reduced to 110 mm after 50 minutes. To find the slope, we use the formula:
Slope = (change in length) / (change in time)
Slope = (110 mm - 150 mm) / (50 minutes - 0 minutes)
Slope = (-40 mm) / (50 minutes)
Slope = -0.8 mm/minute
2) Constructing the linear equation:
We now have the slope (-0.8) and the initial length (150 mm) to create a linear equation:
Length (L) = initial length + slope × time (t)
L = 150 - 0.8t
3) Predicting the time to have 74 mm remaining:
To find the time, plug in 74 mm for the length (L) in the equation and solve for t:
74 = 150 - 0.8t
76 = 0.8t
t = 95 minutes
So, it would take approximately 95 minutes for the wick to have 74 mm remaining.
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If a spinner will land on red 45% of the time, yellow 15% of the time, and blue 40% of the time, what are the chances it wont land on red
Answer:
55%
Step-by-step explanation:
Chances it wont land on red = chances of blue + chances of yellow
= 55%
the human body contains about bacteria.
the human body contains 1 × 1012 about genes. the number of bacteria contained in the human body is how 4 × 104 many times as great as the number of genes contained in the human body?
explain how you arrived at your answer.
The number of bacteria contained in the human body is 40 times as great as the number of genes contained in the human body.
To find out how many times greater the number of bacteria in the human body is than the number of genes, we need to divide the number of bacteria by the number of genes:
4 × 10^13 (number of bacteria) ÷ 1 × 10^12 (number of genes)
= 40
Therefore, the number of bacteria contained in the human body is 40 times as great as the number of genes contained in the human body.
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Prove that the triangle FGH is right-angle des at F
The Pythagorean Theorem is used to prove that triangle FGH is a right triangle at angle F.
What are similar triangles?Similar triangles are triangles that share these two features listed as follows:
Congruent angle measures, as both triangles have the same angle measures.Proportional side lengths, which helps us find the missing side lengths.Considering the equivalent side lengths, the proportional relationship for the side lengths in this problem is given as follows:
4/4.8 = FH/3.6 = 5/6.
Hence the length FH is given as follows:
4/4.8 = FH/3.6
FH = 3.6 x 4/4.8
FH = 3.
If the Pythagorean Theorem is respected, the triangle is a right triangle, hence:
3² + 4² = 5²
9 + 16 = 25
25 = 25. -> right trianglge.
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The highest BASE drop zone in the world is the Kjerag in Norway, where BASE jumpers make an almost straight down plunge at a height of 3,228 feet. The function
represents the time t (in seconds) that it takes a BASE jumper to fall d feet. How far will a BASE jumper fall in 4. 5 seconds?
feet
A BASE jumper will fall 324 feet in 4.5 seconds.
What are velocity ?
velocity is a unit of measurement for the Distance an object travels in a
the predetermined period of time. Here is a word equation that illustrates the connection between space, speed, and time: velocity is calculated by dividing the total Distance traveled by the journey time.
We can use the given function to find out how far a BASE jumper will fall in 4.5 seconds:
d = 16t²
where d is the distance (in feet) and t is the time (in seconds).
Substitute t = 4.5 into the formula:
d = 16(4.5)²
d = 324
Therefore, a BASE jumper will fall 324 feet in 4.5 seconds.
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Let f(x, y)= 1 + 3x² - cos(2y). Find all critical points and classify them as local maxima, local minima, saddle points, or none of these. critical points: (give your answers as a comma separated list of(x, y) coordinates. If your answer includes points that occur at a sequence of values, e.g., at every odd integer, or at any constant multiple of another value, use m for any non-zero even integer, n for any non-zero odd integer, add/or k for other arbitrary constants.) classifications: (give your answers in a comma separated list, specifying maximum, minimum, saddle point, or none for each, in the same order as you entered your critical points)
The critical points and their classifications are: (0, kπ/2), local minimum for all k.
To find the critical points of f(x, y), we need to find where the partial derivatives of f with respect to x and y are equal to zero:
∂f/∂x = 6x = 0
∂f/∂y = 2sin(2y) = 0
From the first equation, we get x = 0, and from the second equation, we get sin(2y) = 0, which has solutions y = kπ/2 for any integer k.
So the critical points are (0, kπ/2) for all integers k.
To classify these critical points, we need to use the second derivative test. The Hessian matrix of f is:
H = [6 0]
[0 -4sin(2y)]
At the critical point (0, kπ/2), the Hessian becomes:
H = [6 0]
[0 0]
The determinant of the Hessian is 0, so we can't use the second derivative test to classify the critical points. Instead, we need to look at the behavior of f in the neighborhood of each critical point.
For any k, we have:
f(0, kπ/2) = 1 + 3(0)² - cos(2kπ) = 2
So all the critical points have the same function value of 2.
To see whether each critical point is a maximum, minimum, or saddle point, we can look at the behavior of f along two perpendicular lines passing through each critical point.
Along the x-axis, we have y = kπ/2, so:
f(x, kπ/2) = 1 + 3x² - cos(2kπ) = 1 + 3x²
This is a parabola opening upwards, so each critical point (0, kπ/2) is a local minimum.
Along the y-axis, we have x = 0, so:
f(0, y) = 1 + 3(0)² - cos(2y) = 2 - cos(2y)
This is a periodic function with period π, and it oscillates between 1 and 3. So for each k, the critical point (0, kπ/2) is neither a maximum nor a minimum, but a saddle point.
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A ninja star, or shuriken, is usually constructed using four congruent isosceles triangles that are placed along the sides of a square. What is the value of x?
a.) 46
b.) 56
c.) 62
d.) 118
The measure of angle x of the ninja star is given by x = 56°
Given data ,
Let the ninja star be represented as an isosceles triangle
Now , it is constructed using four congruent isosceles triangles that are placed along the sides of a square
And , the vertex angle of the triangle is x
In an isosceles triangle, the two equal sides are called legs, and the remaining side is called the base. The angle opposite the base is called the vertex angle
So , the measure of x = 180° - ( 62° + 62° )
On simplifying , we get
x = 56°
Hence , the angle of triangle is x = 56°
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Part B: If GA = 29 and a major arc mDUG = 185°, then determine the minor arc length of
GD.
The length of the minor arc GD is approximately 0.196π units.
To get the length of the minor arc GD, we need to subtract the measure of the major arc mDUG from the circumference of the circle, and then divide by 360° to find the length of one degree of arc.
First, we need to find the circumference of the circle. Since GA = 29, we know that the radius of the circle is also 29. The formula for the circumference of a circle is C = 2πr, so for this circle we have: C = 2π(29) = 58π
Next, we need to subtract the measure of the major arc mDUG from the circumference of the circle. Since mDUG = 185°, we have:
58π - (185/360)(58π) = (175/360)(58π)
Simplifying this expression, we get: (175/360)(58π) = 29(175/72)π ≈ 70.48π
Finally, we divide this value by 360° to find the length of one degree of arc:
(70.48π)/360 ≈ 0.196π
Therefore, the length of the minor arc GD is approximately 0.196π units.
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Use the properties of logarithms to simplify as much as possible. 3) In(4x^5) – In (x^3)- In 4 4) The price of beef has inflated by 2%. If the price of beef inflates 2% compounded biannually, how lung will it take for the price of beef to triple?
3) The expression In(4x^5) - In(x^3) - In 4 can be simplified using the properties of logarithms. We know that ln(a) - ln(b) = ln(a/b) and ln(a^n) = n ln(a), so we can write:In(4x^5) - In(x^3) - In 4 = In[(4x^5)/(x^3)] - In 4= In(4x^2) - In 4= In(4x^2/4)= In(x^2)Thus, the simplified expression is In(x^2).4) To solve this problem, we need to use the formula for compound interest:A = P(1 + r/n)^(nt)where A is the final amount, P is the initial amount, r is the interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years.We want to find t when A = 3P and r = 0.02 (since the price of beef has inflated by 2%). We are told that interest is compounded biannually, so n = 2. Plugging in these values and solving for t, we get:3P = P(1 + 0.02/2)^(2t)3 = (1.01)^2tln(3) = ln(1.01^2t)ln(3) = 2t ln(1.01)t = ln(3) / (2 ln(1.01))Using a calculator, we find t ≈ 34.64 years. Therefore, it will take about 34.64 years for the price of beef to triple at a 2% biannual inflation rate.
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It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.
3) To simplify the expression In(4x^5) - In(x^3) - In(4), we will use the properties of logarithms:
- In(a) - In(b) = In(a/b)
- In(a^b) = b * In(a)
So, we can rewrite the expression as:
In(4x^5 / (x^3 * 4))
Now, we can simplify the expression inside the natural logarithm:
(4x^5) / (4x^3) = x^(5-3) = x^2
Thus, the simplified expression is:
In(x^2)
4) To find how long it will take for the price of beef to triple when inflating 2% compounded biannually, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
where A is the final amount, P is the initial amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. In this case, we want the final amount to be triple the initial amount:
3P = P(1 + 0.02/2)^(2t)
To solve for t, we can divide both sides by P:
3 = (1 + 0.01)^(2t)
Now, take the natural logarithm of both sides and use the properties of logarithms:
ln(3) = ln((1 + 0.01)^(2t))
ln(3) = 2t * ln(1 + 0.01)
Finally, isolate t:
t = ln(3) / (2 * ln(1 + 0.01))
t ≈ 109.96
It will take approximately 110 years for the price of beef to triple when inflating 2% compounded biannually.
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The monthly demand function for product sold by monopoly is p 2,220 1x2 dollars, and the average cost is C = 900 + 14x + x2 dollars. Production is limited to 1,000 units, and x is in hundreds of units_ Find the revenue function, R(x)_ R(x) Find the cost function, C(x): C(x) Find the profit function, P(x) P(x) (a) Find P'(x) . P'(x) Considering the limitations of production, find the quantity (in hundreds of units) that will give the maximum profit. hundred units (b) Find the maximum profit
To find the revenue function, we need to multiply the price (p) by the quantity (x):
R(x) = xp = (2220 - x^2) x
Expanding this expression, we get:
R(x) = 2220x - x^3
To find the cost function, we can simply use the given formula:
C(x) = 900 + 14x + x^2
To find the profit function, we subtract the cost from the revenue:
P(x) = R(x) - C(x)
= (2220x - x^3) - (900 + 14x + x^2)
= -x^3 + 2206x - 900
To find P'(x), the derivative of P(x) with respect to x, we take the derivative of the expression for P(x):
P'(x) = -3x^2 + 2206
Setting P'(x) equal to zero and solving for x, we get:
-3x^2 + 2206 = 0
x^2 = 735.333...
x ≈ 27.104
We can't produce a fraction of a hundred units, so we round down to the nearest hundredth unit, giving x = 27.
To confirm that this value gives a maximum profit, we can check the sign of P''(x), the second derivative of P(x) with respect to x:
P''(x) = -6x
When x = 27, P''(x) is negative, which means that P(x) has a local maximum at x = 27.
Therefore, the quantity that will give the maximum profit is 2700 units (27 x 100).
To find the maximum profit, we evaluate P(x) at x = 27:
P(27) = -(27)^3 + 2206(27) - 900
= 53,955 dollars
Therefore, the maximum profit is $53,955.
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Select the correct answer.
Which is the minimum or maximum value of the given function?
of
44 N₂
O A.
OB.
O.C. The function has a minimum value of -4.
OD. The function has a maximum value of -4.
The function has a minimum value of -3.
The function has a maximum value of -3.
Answer:
C
Step-by-step explanation:
The lowest point on the graph on the y-axis is -4
Evaluate. Assume that x>0. J 563) 8 2 + X X dx
The integral ∫(8x/2 + 2/x^3)dx evaluates to 4x^2 - 2/x^2 + C, where C is the constant of integration.
The given integral ∫(8x/2 + 2/x^3)dx is definite integral without any integration limits. To evaluate this integral, we can split it into two parts
∫8x/2 dx + ∫2/x^3 dx
We made use of the power rule of integration to simplify the first term, and the inverse power rule to simplify the second term.
Simplifying each integral, we get
4x^2 - 2/x^2 + C
where C is the constant of integration.
Therefore, the final answer to the integral is
∫(8x/2 + 2/x^3)dx = 4x^2 - 2/x^2 + C
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--The given question is incomplete, the complete question is given
" Evaluate. Assume that x>0. ∫(8x/2 + 2/x^3)dx"--
Determine the intercepts of the line.
Do not round your answers.
-5x - 4y = 10
Step-by-step explanation:
-5x - 4y = 10
Intercept-y (x = 0)
-5 (0) - 4y = 10
-4y = 10
y = - 5/2
(0, -5/2)
Intercept-x (y = 0)
-5x - 4 (0) = 10
-5x = 10
x = -2
(-2, 0)
#CMIIWQuadrilateral klmn is similar to quadrilateral opqr. find the measure of side op.
round your answer to the nearest tenth if necessary.
n
13
m
r
57
q
27
k
p
0
answer:
submit answer
The measure of side OP in quadrilateral OPQR is 0.
To find the measure of side OP in quadrilateral OPQR, which is similar to quadrilateral KLMN, follow these steps:
1. Identify the corresponding sides in both quadrilaterals. In this case, side OP corresponds to side KL, side OQ corresponds to side KM, side PQ corresponds to side LN, and side QR corresponds to side MN.
2. Determine the scale factor between the quadrilaterals by comparing the lengths of corresponding sides. Since we have the lengths of sides KM (13), LN (27), and MN (57), we can use the ratio of KM/LN (13/27) or MN/LN (57/27) as the scale factor.
3. Apply the scale factor to the length of side KL (0) to find the length of side OP. Since the length of side KL is 0, multiplying by the scale factor (either 13/27 or 57/27) will still result in a length of 0 for side OP.
4. Round your answer to the nearest tenth if necessary. In this case, the length of side OP is 0, so rounding is not necessary.
So, the measure of side OP in quadrilateral OPQR is 0.
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In a group of students,
[tex] \frac{8}{13} [/tex]
of the group are scouts and the rest are police cadets. If
[tex] \frac{3}{8} [/tex]
of the scouts or 15 of them are prefects, calculate the number of police cadets in the group
The calculated number of police cadets in the group is 25
Calculating the number of police cadets in the groupFrom the question, we have the following parameters that can be used in our computation:
Scouts = 8/13
Police cadets = the rest
This means that
Police cadets = 1 - 8/13
Evaluate
Police cadets = 5/13
Also, we have
Prefects = 3/8 or 15 of scouts
This means that
3/8 * Scouts = 15
Scouts = 40
So, we have
Total = 40/(8/13)
Total = 65
Recall that
Police cadets = 5/13
So, we have
Police cadets = 5/13 * 65
Evaluate
Police cadets = 25
Hence, the number of police cadets in the group is 25
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Subtract 1/4- 5/14 = type an integer or fraction
Answer:
-3/28
Step-by-step explanation:
In order to subtract fractions you first have to get a common denominator. The common denominator between 4 and 14 is 28 since 4(7) = 28 and 14(2) = 28. Then multiply the top and bottom of (1/4) by (7/7) and (5/14) by (2/2). This gives a common denominator and since you multiply by a fraction equal to 1, it doesn't change the value. Therefore, we get 7/28 - 10/28 = -3/28.
if integrate f(x) dx = 1/3 * x ^ 3 - 2x ^ 2 + 3x - 5 then the value of f(-2) = ...
A. -9
B. -1
C. 7
D. 15
E. 23
please
pleasee
The value of f(-2) is 15.
Hence option D is correct.
The given expression is
[tex]\int\limits {f(x)} \, dx[/tex] = (1/3)x³ - 2x² +3x - 5
To find expression of f(x)
Differentiate it with respect to x
f(x) = x² - 4x + 3
Now put x = -2
f(-2) = (-2)² - 4(-2) + 3
= 4 + 8 + 3
= 15
Hence,
f(-2) = 15
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what are the values of m and n, and what does the plotted graph look like?
The values of m and n are
m = 0 and n = 3.125
The graph is attached
How to find the values of m and nThe values of m and n are solved using the relationship between miles and kilometers. This type of relationship is a linear proportional relationship. Linear relationship implies the graph will be a straight line graph.
This relationship is that 1 mile equals 0.625 km hence the linear equation is
y = 0.625x
when x = 0, we have that
y = 0.625 * 0
y = 0
when x = 5, we have that
y = 0.625 * 5
y = 3.125
Where x is kilometers and y is miles
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Kai bought 5 bags. In each bag there is bottle of Gatorade that cost 3$ and two pacIs of gum. If Kai spent 55$ all together how much did each pack of gum cost?
If Kai spent 55$ all together then each pack of gum cost $4.
To solve this question follow the steps given below:
Calculate the total cost of Gatorade.
Since there are 5 bags and each bag has a bottle of Gatorade that costs $3, the total cost for Gatorade is 5 * $3 = $15.
Calculate the total cost of gum.
Since Kai spent $55 in total, we need to subtract the cost of Gatorade to find the total cost of gum. $55 - $15 = $40.
Calculate the total number of gum packs.
Each bag contains 2 packs of gum, and there are 5 bags. So, there are 2 * 5 = 10 packs of gum.
Calculate the cost of each pack of gum.
To find the cost of each pack of gum, divide the total cost of gum by the number of gum packs. $40 / 10 = $4.
So, each pack of gum cost $4.
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What is the exact value of sin−1(−12)? Enter your answer in the box. Sin−1(−12) = 1$$ Correct answers: 1−π6
The exact value of sin⁻¹(−1/2) is -π/6.
Given, sin⁻¹(-1/2)
The inverse sine function, sin⁻¹, or arcsin, returns the angle whose sine is equal to the given value. In this case, we are looking for the angle whose sine is -1/2.
Let y = sin⁻¹(-1/2)
sin (y) = -1/2
sin (y) = - sin (π/6)
sin (y) = sin (- π/6)
y = - π/6
sin⁻¹(-1/2) = - π/6
To understand why the answer is -π/6, we can consider the unit circle. On the unit circle, the sine function represents the y-coordinate of a point corresponding to an angle. For -1/2, we need to find the angle where the y-coordinate is -1/2.
One such angle is -π/6, where the point on the unit circle is located in the fourth quadrant. At this angle, the y-coordinate is -1/2. Hence, sin⁻¹(−1/2) is -π/6.
Therefore, the exact value of sin⁻¹(−1/2) is -π/6.
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the top of the farm silo is a hemisphere with a radius of 9ft. the bottom of the silo is a cylinder with a height of 35ft. how many cubic feet of grain can the solo hold? use 3.14 for pi and round your answer to the nearest cubic foot.
To find the total volume of the silo, we need to add the volume of the hemisphere on top to the volume of the cylinder at the bottom.
The volume of a hemisphere is given by:
V_hemi = (2/3)πr^3
where r is the radius of the hemisphere.
Substituting r = 9ft, we get:
V_hemi = (2/3)π(9ft)^3
= 1521π ft^3
The volume of a cylinder is given by:
V_cyl = πr^2h
where r is the radius of the cylinder and h is its height.
Substituting r = 9ft and h = 35ft, we get:
V_cyl = π(9ft)^2(35ft)
= 2673π ft^3
Therefore, the total volume of the silo is:
V_silo = V_hemi + V_cyl
= 1521π + 2673π
= 4194π ft^3
≈ 13160 ft^3
Rounding to the nearest cubic foot, the silo can hold approximately 13160 cubic feet of grain.
Which equation best represents the line of best fit for the scatterplot?a) y = 2.5x + 25 b) y = −2.5x + 20 c) y = −0.05x + 20 d) y = −0.005x + 22.5
The equation of the line representing that best fit the given scatterplot is given by option d. y = -0.005x + 22.5.
Consider the two points from the attached scatterplot.
Let the coordinates of the two point be ( x₁ , y₁) = ( 1500 , 12.5 )
And other point be ( x₂ , y₂) = ( 2000 , 10 )
Slope of the line 'm' = ( y₂ - y₁ ) / ( x₂ - x₁ )
= ( 10 - 12.5) / ( 2000 - 1500 )
= -2.5 / 500
= -0.005
From the attached scatterplot we have,
y-intercept 'c' where x = 0 is equals to 22.5.
The equation best which represents the line of best fit for the scatterplot is equals to,
y = mx + c
Substitute the value we have,
y = -0.005x + 22.5
Therefore, the equation of the line representing scatterplot is equals to option d. y = -0.005x + 22.5.
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The above question is incomplete, the complete question is:
Which equation best represents the line of best fit for the scatterplot?
a) y = 2.5x + 25 b) y = −2.5x + 20 c) y = −0.05x + 20 d) y = −0.005x + 22.5
Attached scatterplot.
Convert the rectangular coordinates (0, 6√3) into polar form. Express the angle using radians in terms of over the interval 0 ≤ 0 < 27, with a positive value of r.
Answer:
The polar form of the rectangular coordinates (0, 6√3) with a positive value of r, over the interval 0 ≤ θo < 27 and in terms of radians, is (6√3, 1.58).
Step-by-step explanation:
To convert the rectangular coordinates (0, 6√3) to polar form, we can use the following formulas:
r = √(x^2 + y^2)
θ = tan^(-1)(y/x)
Substituting the given values, we get:
r = √(0^2 + (6√3)^2) = 6√3
θ = tan^(-1)((6√3)/0) = π/2
However, note that the angle θ is not well-defined since x=0. We can specify that the point lies on the positive y-axis, which corresponds to θ = π/2 radians.
Thus, the polar form of the rectangular coordinates (0, 6√3) is:
r = 6√3
θ = π/2
To express the angle θ in terms of θo, where 0 ≤ θo < 27 and in radians, we can write:
θ = π/2 = (π/54) × 54 ≈ (0.0292) × 54 ≈ 1.58 radians
Therefore, the polar form of the rectangular coordinates (0, 6√3) with a positive value of r, over the interval 0 ≤ θo < 27 and in terms of radians, is (6√3, 1.58).
4 3 (1)/(5 )2 (3)/(5 )1 (4)/(5)
ecplict formula, in slope intercept form (4)/(5)
The explict formula, in slope intercept form is an = n/5
Calculating the explict formula, in slope intercept formThe given sequence is 1/5, 2/5, 3/5.
We can observe that this is an arithmetic sequence, where the first term is 1/5, the common difference is 1/5
To find the explicit formula for an arithmetic sequence, we can use the formula:
an = a1 + (n-1)d
Substituting the values we know for this sequence, we get:
an = 1/5 + (n - 1)*(1/5)
Evaluate
an = n/5
Thus, the nth term of this sequence can be found by substituting the value of n in the formula an = n/5
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Complete question
1/5 2/5 3/5
What is the explicit formula in slope intercept form
Your doing practice 5
5.) The dimensions of the banner whose perimeter is given is listed below:
length = 134in
width = -59in
How to determine the dimensions of the rectangular banners?To calculate the dimensions of the rectangular banner, the formula for the perimeter of rectangle should be used and it's given below;
Perimeter of rectangle = 2(length+width)
length = 16-2a
width = a
perimeter = 160in
That is;
160 = 2(16-2a+a)
160 = 32-4a+2a
160 = 42-2a
2a = 42-160
2a = -118
a = 118/2
= -59
Length = 16-2(-59)
= 16+118
= 134in
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HELP PLS & THANK YOU
Answer:
Step-by-step explanation:
Lola is in a hot air balloon that has just taken off and is now floating 7 meters above its launching point. Julian is standing on the ground, 6 meters away from the launching point. How far apart are Lola and Julian? If necessary, round to the nearest tenth.
It costs $ 35 per hour to rent a boat at the lake. You also need to pay a $ 25 fee for safety equipment. You have $ 200 . How long can you rent the boat? CLEAR CHECK Write the equation that represents the situation. 35 + = $ 200 Solve for h . h = hours
Answer: H = 5
the equation would be, 35h - 25 = $200
Step-by-step explanation:
Use Mean value theorem to prove √6a + 3 < a + 2 for all a > 1.
Using methods other than the Mean Value Theorem will yield no marks. {Show all reasoning).
√6a + 3 < a + 2, we have proven that √6a + 3 < a + 2 for all a > 1 using the Mean Value Theorem.
To use the Mean Value Theorem to prove √6a + 3 < a + 2 for all a > 1, we first note that the function f(x) = √6x + 3 is continuous on the interval [1,a].
Next, we need to find a point c in the interval (1,a) such that the slope of the line connecting (1,f(1)) and (a,f(a)) is equal to the slope of the tangent line to f(x) at c.
The slope of the line connecting (1,f(1)) and (a,f(a)) is given by:
(f(a) - f(1)) / (a - 1) = (√6a + 3 - √9) / (a - 1) = (√6a) / (a - 1)
To find the slope of the tangent line to f(x) at c, we first find the derivative of f(x):
f'(x) = (1/2) * (6x + 3)^(-1/2) * 6 = 3 / √(6x + 3)
Then, we evaluate f'(c) to get the slope of the tangent line at c:
f'(c) = 3 / √(6c + 3)
Now, by the Mean Value Theorem, there exists a point c in (1,a) such that:
f'(c) = (√6a) / (a - 1)
Setting these two expressions for f'(c) equal to each other, we get:
3 / √(6c + 3) = (√6a) / (a - 1)
Solving for c, we get:
c = (a + 2) / 6
(Note that c is indeed in (1,a) since a > 1.)
Now, we can evaluate f(a) and f(1) and use the Mean Value Theorem to show that:
√6a + 3 - √9 < (√6a) / (a - 1) * (a - 1)
Simplifying, we get:
√6a + 3 < a + 2
as desired.
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11. The volume of a cuboid with a square base is given 5 by (2x¹ + xy-2y) m². 5 (i) Factorise the expression 2x² + xy-2y². 1 (ii) The cuboid has a height of m. Given that the length of each side of the base can be expressed as (px - qy) m or (qx + py) m, using your answer from part (i), state the value of p and of q. (iii) Hence, express x in terms of y.
A website’s profit as a function of visitors is represented in the table. the function is quadratic.
visitors (thousands). monthly profit ($ thousands)
0. 0
0. 5. −10
1. 0
2. 80
3. 240
select from the drop-down menu to correctly complete the sentence.
the y-intercept represents
options:
the number of visitors when the publisher breaks even
the maximum amount of profits
the amount of profit when there are 0 visitors
The amount of profit when there are 0 visitors.
In the context of the given quadratic function, the y-intercept represents the amount of profit when there are 0 visitors.
To explain, the y-intercept is the point at which the function intersects the y-axis. This occurs when the x-value (number of visitors in thousands) is 0. In the given table, the y-intercept corresponds to the data point (0, 0), meaning there is a profit of $0 thousands when there are 0 visitors.
Looking at the table, we can see that when the number of visitors is 0, the profit is also 0. Therefore, the y-intercept of the function represents the amount of profit when there are 0 visitors.
So the correct option is: "the amount of profit when there are 0 visitors".
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