a) There are 1540 ways that the teacher can select the three students.
b) The probability that all three students selected are girls is approximately 0.176 or 17.6%.
c) The probability that at least one boy is selected is approximately 0.824 or 82.4%.
a)
To find the number of ways the teacher can select three students out of 22 students (7 boys and 15 girls), we can use the combination formula. The number of ways to select r items from a set of n items is given by:
nCr = n! / (r! * (n-r)!)
where n! represents the factorial of n (i.e., n! = n x (n-1) x (n-2) x ... x 3 x 2 x 1), and r! represents the factorial of r. Applying this formula, we get:
22C3 = 22! / (3! * (22-3)!) = 22! / (3! * 19!) = (22 x 21 x 20) / (3 x 2 x 1) = 1540
Therefore, there are 1540 ways that the teacher can select the three students.
b)
To find the probability that all three students selected are girls, we can use the formula for the probability of an event occurring. Since there are 15 girls and 7 boys, the probability of selecting a girl is 15/22 for the first selection, 14/21 for the second selection (since there are now 14 girls left out of 21 remaining students), and 13/20 for the third selection. Applying the formula, we get:
P(all three are girls) = (15/22) x (14/21) x (13/20) ≈ 0.176
Therefore, the probability that all three students selected are girls is approximately 0.176 or 17.6%.
c)
To find the probability that at least one boy is selected, we can use the complement rule. The complement of selecting at least one boy is selecting all three girls, which we calculated in part (b) to be approximately 0.176. Therefore, the probability of selecting at least one boy is:
P(at least one boy) = 1 - P(all three are girls) ≈ 1 - 0.176 ≈ 0.824
Therefore, the probability that at least one boy is selected is approximately 0.824 or 82.4%.
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Select the correct answer
a mine extracts 2 metric tons of coal in an hour. the
number of hours spent mìning, which expression re
oa. the expression is at. the amount of ore
ob. the expression
The expression that represents the amount of ore sold and how much ore can the mine sell after extracting ore for 12 hours is option B: The expression is 2t−14t. The amount of ore is 21 metric tons.
The reasoning for the selection of the expression and amount of ore can the mine sell after extracting ore for 12 hours is as follows.
1: Determine the amount of coal used for electricity generation in terms of t.
The mine uses 14 tons of coal every hour, so the total amount used for electricity generation is 14t.
2: Determine the total amount of coal extracted in terms of t.
The mine extracts 2 tons of coal every hour, so the total amount extracted is 2t.
3: Calculate the amount of coal sold in terms of t.
To find the amount of coal sold, subtract the amount used for electricity generation from the total amount extracted: 2t - 14t.
4: Determine the amount of coal sold after 12 hours.
Substitute t = 12 into the expression:
2(12) - 14(12) = 24 - 168 = -144.
However, since the mine uses 14 tons of the extracted coal every hour, it cannot sell more coal than it extracts. So, the correct expression should be 2t - 14 (without the t for the amount used for electricity generation).
5: Calculate the amount of coal sold after 12 hours using the corrected expression.
Substitute t = 12 into the expression: 2(12) - 14 = 24 - 14 = 10 metric tons.
The correct expression should be 2t - 14, and the amount of coal the mine can sell after extracting coal for 12 hours is 10 metric tons. Hence, the correct answer is option B.
Note: The question is incomplete. The complete question probably is: A mine extracts 2 metric tons of coal in an hour. The mine uses 14 ton of the extracted coal every hour to generate electricity for the mine and sells the rest. If t is the number of hours spent mining, which expression represents the amount of ore sold? How much ore can the mine sell after extracting ore for 12 hours? A) The expression is 2t−1/4t. The amount of ore is 23 3/4 metric tons. B) The expression is 2t−1/4t. The amount of ore is 21 metric tons. C) The expression is 2t+1/4t. The amount of ore is 24 metric tons. D) The expression is 2t+1/4t. The amount of ore is 24 1/4 metric tons.
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The regular price of a red T-shirt is $6.93. Ernest has a coupon for $6.75 off. How much will Ernest pay for the T-shirt?
Answer:
18 cent
Step-by-step explanation:
You get a job as a nurse. Your salary for the first year is $74,000. You will
receive a 1.2% increase every year. If you could save your entire salary, how
much money would you have in 5 years? Round to the nearest cent (2 decimal
places). Hint: What is a₁? What is r? Then use the formula for a finite
geometric series.
Answer: The amount of money you would have in 5 years if you could save your entire salary with a 1.2% increase every year would be $87,357.41.
Explanation:
The initial term, a₁, is $74,000, and the common ratio, r, is 1 + 1.2% = 1.012. To find the sum of the first 5 terms, we use the formula for a finite geometric series:
S₅ = a₁(1 - r⁶)/(1 - r)
Plugging in the values, we get:
S₅ = $74,000(1 - 1.012⁵)/(1 - 1.012) = $87,357.41 (rounded to the nearest cent)
Therefore, if you save your entire salary, you would have approximately $87,357.41 in 5 years with a 1.2% increase every year.
Find the point (s) on the curve y = x^2/6 closest to the point (0,0) The points) are
The point(s) on the curve y = x²/6 closest to the point (0,0) are (0,0) and (±√2, 2/3).
To find the point(s) on the curve y = x²/6 closest to the point (0,0), we can use the distance formula between two points:
d = √((x₁ - x₂)² + (y₁ - y₂)²)
where (x₁, y₁) is a point on the curve and (x₂, y₂) is the point (0,0).
We want to minimize the distance d, which is equivalent to minimizing d². Therefore, we can minimize:
d² = (x₁ - 0)² + (y₁ - 0)²
= x₁² + y₁²
subject to the constraint that the point (x₁, y₁) is on the curve y = x²/6.
Substituting y = x²/6 into the expression for d², we get:
d² = x₁² + (x₁²/6)
= (7/6)x₁²
To minimize d², we minimize x₁². Since x₁² is always non-negative, the minimum occurs when x₁² = 0 or when the derivative of d² with respect to x₁ is zero.
Taking the derivative of d² with respect to x₁, we get:
d²/dx₁ = (7/3)x₁
Setting this equal to zero, we get x₁ = 0.
Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).
To find the other closest point(s), we can solve y = x²/6 for x² and substitute it into the expression for d²:
x² = 6y
d² = 7x²/6 = 7y
Therefore, to minimize d², we need to minimize y. Since y is always non-negative, the minimum occurs when y = 0 or when the derivative of d² with respect to y is zero.
Taking the derivative of d² with respect to y, we get:
d²/dy = 7
Setting this equal to zero, we get y = 0.
Substituting y = 0 into y = x²/6, we get x = 0. Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).
To find the other closest point, we can solve y = x²/6 for x:
x² = 6y
x = ±√(6y)
Substituting this into the equation for y, we get:
y = (√(6y))²/6 = 2/3
Therefore, the other closest points are (±√2, 2/3).
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The circumference (C) of a circle is 16 cm. Which formula can you use to find the diameter (d) if you know that C = π
d?
Answer:
c/π=d
explanation:
d × π = c
divide c to isolate d
Answer: I would multiply pie by a diameter until it equals 16.
(I know this probably isn’t the professional way but it should work.
The greenery landscaping company puts in an order for 2 pine trees and 5 hydrangea bushes for a neighborhood project. the order costs $150. they put in a second order for 3 pine trees and 4 hydrangea bushes that cost $144.50.
2(2.5) + 5(23) = 150
3(2.5) + 4(23) = 144.5
Both equations are satisfied, so our solution is correct.
The greenery landscaping company orders how many trees and bushes for the neighborhood?To solve the problem, let's first assign some variables. Let x be the cost of one pine tree and y be the cost of one hydrangea bush. We can then use these variables to set up a system of equations:
2x + 5y = 150 (equation 1)
3x + 4y = 144.5 (equation 2)
We can solve this system of equations using various methods. Here, we will use the substitution method.
From equation 1, we can solve for x in terms of y:
2x = 150 - 5y
x = (150 - 5y)/2
We can then substitute this expression for x into equation 2:
3((150 - 5y)/2) + 4y = 144.5
Multiplying both sides by 2 to eliminate the fraction:
3(150 - 5y) + 8y = 289
Expanding and simplifying:
450 - 15y + 8y = 289
-7y = -161
y = 23
We can now substitute this value for y into either equation 1 or 2 to solve for x:
2x + 5(23) = 150
2x = 5
x = 2.5
Therefore, one pine tree costs $2.50 and one hydrangea bush costs $23.
To check our work, we can substitute these values into both equations:
2(2.5) + 5(23) = 150
3(2.5) + 4(23) = 144.5
Both equations are satisfied, so our solution is correct.
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HELPPP SOMEBODY PLEASEEE WITH THIS MATHHHH
The correct statement is given as follows:
The function g(t) reveals the market value of the house increases by 3.6% each year.
How to define an exponential function?An exponential function has the definition presented as follows:
[tex]y = ab^x[/tex]
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.The parameter b for this problem is given as follows:
b = 1.036.
As the parameter b has an absolute value greater than 1, the function is increasing, with a rate given as follows:
1.036 - 1 = 0.036 = 3.6% a year.
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The measures of the angles of a triangle are shown in the figure below. Solve for x.
(2x+16) 48degrees
Answer:
x = 16
Step-by-step explanation:
(2x + 16) = 48
Subtract 16 with the positive 16 to cancel the numbers.
Subtract 16 with 48.
2x = 32
divide 32 by 2 to isolate the x.
32/2 = 16
x = 16
Find the length of the segment indicated. Round your answer to the nearest tenth if necessary.
The value of x in the given circle is 18.1 units.
Given is a circle, where two radii are given one chord is given,
We need to find the value of the x which is also the radius,
We know all the radii in a circle are equal,
So, here the radius = 7.9+10.2 = 18.1 units.
Hence the value of x in the given circle is 18.1 units.
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(3X-5)^1/4+3=4
Your anwser should be x=2!
SHOW WORK
(Explanation below)
x=2
x = 2 is the solution of the equation
What is Equation?Two or more expressions with an Equal sign is called as Equation.
The given equation is [tex](3X-5)^(^1^/^4^) + 3 = 4[/tex]
We have to find the value of x
Subtracting 3 from both sides:
[tex](3X-5)^(^1^/^4^) = 1[/tex]
Raising both sides to the fourth power:
3X - 5 = 1^4
3X - 5 = 1
Adding 5 to both sides:
3X = 6
Dividing by 3:
X = 2
Therefore, x = 2 is the solution of the equation
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compute (7 4/9 -8)*3.6-1.6*(1/3-3/4)+ 1 2/5 ÷(0.35)
The value of (7 4/9 -8)*3.6-1.6*(1/3-3/4)+ 1 2/5 ÷(0.35) is given as 241/54.
How to solve for the value(7 4/9 -8) = -5/9.
3.6-1.6 = 2.0
1/3-3/4 = 1/3 - 3/4
= 4/12 - 9/12
= -5/12
we will have -5/9 * 2 = -10/9.
-10/9 * -5/12
10/9 * -5/12 = (10 * 5) / (9 * 12) = 50/108
We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 2:
50/108 = 25/54
we will have
25/54 + 1 2/5 ÷(0.35)
1 2/5 ÷ 0.35 = (7/5) ÷ (35/100) = (7/5) * (100/35) = 4
Now, we can substitute this value into the expression:
25/54 + 4 = (25/54) + (216/54) = 241/54
Therefore, the value of the expression 25/54 + 1 2/5 ÷(0.35) is 241/54.
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20
Sean pays £10 for 24 chocolate bars.
He sells all 24 chocolate bars for 50p each.
Work out Sean's percentage profit. .
Sean's percentage profit is 20% on selling 24 chocolate bars.
What is Sean's percentage profit?
Sean's cost price for each chocolate bar is:
£10 / 24 bars = £0.4167 per bar
Sean sells each chocolate bar for 50p, which is £0.5
Sean's revenue from selling all 24 chocolate bars is:
24 bars x £0.5 per bar = £12
Sean's profit is the difference between his revenue and his cost:
Profit = £12 - £10 = £2
To calculate the percentage profit, we can use the following formula:
Percentage profit = (Profit / Cost price) x 100%
So, plugging in the values we get:
Percentage profit =[tex](2 / 10) x 100% = 20%[/tex]= 20
Therefore, Sean's percentage profit is 20%. He earned a profit of £2 on his initial investment of £10, which is equivalent to a 20% return on investment.
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You can find the area of a trapezoid by decomposing it into a rectangle and one or more triangles you can find the area of a kite by decomposing it into triangles
The statement on finding the areas of a trapezoid and a kite are True.
How to find area by decomposing shapes ?To determine the area of a trapezoid, it can be broken down into separate geometrical shapes. One possible breakdown would include a rectangle with two adjacent right triangles or an isosceles triangle with one right triangle configuration. By calculating each smaller compartment's size and summing them together, one can obtain the total area for the trapezoid.
Similarly, in order to find the surface area of a kite shape, drawing a diagonal creates two adjoining triangles that are easily computed individually then summed.
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Options for this question :
True
False
A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options. m∠5 + m∠3 = m∠4 m∠3 + m∠4 + m∠5 = 180° m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6 m∠2 + m∠3 + m∠5 = 180°
The true triangle statement regarding the diagram are:
1. m∠5 + m∠6 = 180° ________Linear Pair
2. ∠ 2+ ∠ 3 = ∠ 6________Exterior angle Property of Triangle
3. m∠2 + m∠3 + m∠5 = 180°________Triangle Sum Property
What is the angle measurement?From the question, Δ ABC with Exterior angles as ∠ 1 , ∠ 4 ,and ∠ 6
Note that the Exterior angle Property of Triangle state that An exterior angle of a triangle is equal to the sum of the opposite interior angles.
Hence: For Exterior ∠ 1 :
∠ 1 = ∠ 5 + ∠ 3 ________Exterior angle Property of Triangle
Also,
For Exterior ∠ 4:
∠ 4 = ∠ 5 + ∠ 2 ________Exterior angle Property of Triangle
Also,
In regards to Exterior ∠ 6:
∠ 6 = ∠ 2 + ∠ 3 ________ Exterior angle Property of Triangle
Using Triangle Sum Property, it state that In a triangle sum of the measures of angles is equal to 180° Hence: m∠2 + m∠3 + m∠5 = 180° ________Triangle Sum Property
The Linear Pair will be: The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.
Therefore, m∠5 + m∠6 = 180° ________Linear Pair
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See full question below
A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options.
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 =
m∠6 m∠2 + m∠3 + m∠5 = 180°
(a) What is the mean of this stem and leaf plot? Show your work. What is the median of the data? Show your work
The mean of the given stem and leaf plot is 24.5 and the median of the data is 25.
The stem and leaf plot represents the given data as:
| 2 | 4, 5, 6, 9
| 3 | 1, 4, 5, 5, 7, 8
| 4 | 2, 5, 7, 8, 9
To find the mean, we need to add up all the values and divide by the total number of values.
Mean = (24 + 25 + 26 + 29 + 31 + 34 + 35 + 35 + 37 + 38 + 42 + 45 + 47 + 48 + 49) / 15
= 367 / 15
= 24.5
To find the median, we need to arrange the data in order and find the middle value. As there are 15 data points, the median will be the average of the 8th and 9th data points.
Data in order: 24, 25, 25, 26, 29, 31, 34, 35, 35, 37, 38, 42, 45, 47, 48
Median = (35 + 37) / 2
= 36.
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Kali has a choice of 20 flavors for her triple scoop cone. If she
chooses the flavors at random, what is the probability that the 3 flavors she
chooses will be vanilla, chocolate, and strawberry?
Find the inverse for each relation: 4 points each
1. {(1,‐2), (2, 3),(3, ‐3),(4, 2)}
2. {(4,2),(5,1),(6,0),(7,‐1)}
Find an equation for the inverse for each of the following relations.
3. Y=-8x+3
4. Y=2/3x-5
5. Y=1/2x+10
6. Y=(x-3)^2
Verify that f and g are inverse functions.
7. F(x)=5x+2;g(x)=(x-2)/5
8. F(x)=1/2x-7;g(x)=2x+14
The inverse for each relation:
1. {(1,‐2), (2, 3),(3, ‐3),(4, 2)} - {(-2, 1), (3, 2), (-3, 3), (2, 4)}
2. {(4,2),(5,1),(6,0),(7,‐1)} - {(2, 4), (1, 5), (0, 6), (-1, 7)}
3. Inverse equation: y=(-1/8)x+3/8
4. Inverse equation: y=3/2x+15/2
5. Inverse equation: y=2x-20
6. Inverse equation: y=[tex]x^{(1/2)}+3[/tex]
7. Since fog(x) = gof(x) = x, f and g are inverse functions.
8. Since fog(x) = gof(x) = x, f and g are inverse functions.
1. To find the inverse of the relation, we need to swap the positions of x and y for each point and then solve for y.
{(1, -2), (2, 3), (3, -3), (4, 2)}
Inverse: {(-2, 1), (3, 2), (-3, 3), (2, 4)}
2. Again, we swap x and y and solve for y.
{(4, 2), (5, 1), (6, 0), (7, -1)}
Inverse: {(2, 4), (1, 5), (0, 6), (-1, 7)}
3. To find the inverse equation for y=-8x+3, we swap x and y and solve for y.
x=-8y+3
x-3=-8y
y=(x-3)/-8
Inverse equation: y=(-1/8)x+3/8
4. To find the inverse equation for y=2/3x-5, we swap x and y and solve for y.
x=2/3y-5
x+5=2/3y
y=3/2(x+5)
Inverse equation: y=3/2x+15/2
5. To find the inverse equation for y=1/2x+10, we swap x and y and solve for y.
x=1/2y+10
x-10=1/2y
y=2(x-10)
Inverse equation: y=2x-20
6. To find the inverse equation for y=(x-3)², we swap x and y and solve for y.
x=(y-3)²
[tex]x^{(1/2)}=y-3[/tex]
[tex]y=x^{(1/2)}+3[/tex]
Inverse equation: [tex]y=x^{(1/2)}+3[/tex]
7. To verify that f(x)=5x+2 and g(x)=(x-2)/5 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.
fog(x) = f(g(x)) = f((x-2)/5) = 5((x-2)/5) + 2 = x
gof(x) = g(f(x)) = g(5x+2) = ((5x+2)-2)/5 = x/5
Since fog(x) = gof(x) = x, f and g are inverse functions.
8. To verify that f(x)=1/2x-7 and g(x)=2x+14 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.
fog(x) = f(g(x)) = f(2x+14) = 1/2(2x+14) - 7 = x
gof(x) = g(f(x)) = g(1/2x-7) = 2(1/2x-7) + 14 = x
Since fog(x) = gof(x) = x, f and g are inverse functions.
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Ms. Regan is making a circular quilt and wants to include a lace pattern
around the outside of the quilt. If the area of the quilt is 28. 26 square feet, how many feet of lace does Ms. Regan need to purchase? (Use 3. 14 for pi. )
To find out how much lace Ms. Regan needs to purchase, we first need to calculate the circumference of the circular quilt. We know that the area of the quilt is 28.26 square feet, and we can use the formula A = πr^2 to find the radius of the quilt.
28.26 = 3.14 x r^2
r^2 = 9
r = 3
Now that we know the radius is 3 feet, we can use the formula C = 2πr to find the circumference of the quilt.
C = 2 x 3.14 x 3
C = 18.84 feet
Therefore, Ms. Regan needs to purchase 18.84 feet of lace to go around the outside of her circular quilt.
In summary, to find out how much lace Ms. Regan needs to purchase, we need to calculate the circumference of the circular quilt. We do this by first finding the radius using the formula A = πr^2. Once we know the radius, we can use the formula C = 2πr to find the circumference. In this case, the circumference is 18.84 feet, so Ms. Regan needs to purchase that amount of lace.
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CAN SOMEONE SHOW ME STEP BY STEP ON HOW TO DO THIS
A city just opened a new playground for children in the community. An image of the land that the playground is on is shown.
A polygon with a horizontal top side labeled 45 yards. The left vertical side is 20 yards. There is a dashed vertical line segment drawn from the right vertex of the top to the bottom right vertex. There is a dashed horizontal line from the bottom left vertex to the dashed vertical, leaving the length from that intersection to the bottom right vertex as 10 yards. There is another dashed horizontal line that comes from the vertex on the right that intersects the vertical dashed line, and it is labeled 12 yards.
What is the area of the playground?
900 square yards
855 square yards
1,710 square yards
This graph represents the equation y=(x-5)^2-1 .
How many ordered pairs (x, y) for 3 < x < 7 satisfy this equation?
There are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.
To find the ordered pairs (x, y) for 3 < x < 7 that satisfy the equation y=(x-5)^2-1, follow these steps:
Step 1: Set the range of x values: 3 < x < 7
Step 2: Plug in each whole number value of x within the given range (4, 5, and 6) into the equation and calculate the corresponding y values.
For x = 4:
y = (4 - 5)^2 - 1
y = (-1)^2 - 1
y = 0
For x = 5:
y = (5 - 5)^2 - 1
y = (0)^2 - 1
y = -1
For x = 6:
y = (6 - 5)^2 - 1
y = (1)^2 - 1
y = 0
Step 3: Write the ordered pairs (x, y) based on the calculated y values.
For x = 4, the ordered pair is (4, 0)
For x = 5, the ordered pair is (5, -1)
For x = 6, the ordered pair is (6, 0)
In the given range, there are 3 ordered pairs (x, y) that satisfy the equation y=(x-5)^2-1.
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A father and his three children decide on all matters with a vote. Each member of the family gets as many votes as their age. Right now, the family members are 36, 13, 6, and 4 years old, so the father always wins. How many years will it take for the three children to win a vote if they all agree? Show your work.
Answer:
Step-by-step explanation:
Answer:
13 years
Step-by-step explanation:
Intuition for how sons can collectively win after a certain period of time:- After a certain period of time the father's age will increase by that certain period of time (say 5 years) but for the sons (since there are 3 of them) their collective age will increase by three times that of their father (5 for each 1 one them). Therefore there exist a time after which collective increase in sons' age can cover the current gap of 13 years.
In January, 280 guests at a hotel chose to use the valet service to park their cars during their stay. At the same time, 120 guests chose to use a public parking garage for their cars during their stay. What percentage of the guests at this hotel used the valet service?
70 percent of the guests at this hotel used the valet service.
To find the percentage of guests who used the valet service, we can follow these steps:
1. Add the number of guests who used the valet service (280) and those who used the public parking garage (120) to find the total number of guests with cars: 280 + 120 = 400 guests.
2. Divide the number of guests who used the valet service (280) by the total number of guests with cars (400).
3. Multiply the result by 100 to convert it into a percentage.
So, let's calculate the percentage:
(280 / 400) * 100 = 0.7 * 100 = 70%
Thus, 70% of the guests at this hotel used the valet service.
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Matemáticamente estos dos conjuntos son lo mismo o son una manera de reescribir al otro o son distintos? (2,6) y [1,5]
The sets (2,6) and [1,5] are not the same mathematically but they do have some overlap.
What is the text about?The first pair, (2,6), signifies a number line interval that is open and commences at 2, concluding at 6, while excluding the endpoints.
So one can say that the closed interval on the number line between 1 and 5, including both endpoints, is represented by the set [1,5]. any integer that is seen between 1 and 5, inclusive, is included in this set.
Although there is some similarity between the two groups, namely the presence of numbers 2 to 5, they are distinct from each other. The numerical interval (2,6) does not contain the values 2 and 6, whereas those two numbers are part of the range [1,5].
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Mathematically these two sets are the same or are they a way of rewriting the other or are they different? (2,6) and [1,5]
measured in astronomical units, can be modeled using the expression ((1)/(52)x)^((2)/(3)) , where x is the number of Earth weeks it takes for the planet to orbit the sun. Which expression could also be used to represent the average distance of a planet from the sun using radicals?
So the expression that represents the average distance of a planet from the sun using radicals is: d = k/2√13 * √x
What is exponent?An exponent, also known as a power, is a mathematical notation that indicates the number of times a quantity is multiplied by itself. It is usually written as a small number (the exponent) placed to the right and above a larger number (the base). Exponents are used in many mathematical concepts, including logarithms, roots, and scientific notation.
Here,
The expression ((1)/∛(52)x)²) can be simplified using exponent rules:
((1)/∛(52)x)²) =((1)/∛(52)x)²) * ∛x²)
= 1/(∛52² * ∛x²)
The average distance of a planet from the sun measured in astronomical units can be represented using the formula:
d = k * √T
where d is the distance from the sun, T is the time it takes for the planet to orbit the sun, and k is a constant of proportionality.
We can rewrite this formula in terms of Earth weeks by noting that there are 52 weeks in a year, so T = (1/52)x years. Substituting this into the formula, we get:
d = k * √((1/52)x)
Simplifying this expression using exponent rules, we get:
d = k * √(1/52)* √x
So an equivalent expression using radicals to represent the average distance of a planet from the sun is:
d = k * √(1/(52)) * √x
which simplifies to:
d = k/√(52) * √x
or
d = k/2√13 * √x
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i need help its due in 2 hours
Answer:
C. The product of two irrational numbers is irrational.
Example: √3•√3=3
Question 9 Previous Consider the indefinite integral 63% (6x3 + 10x2 + 64x + 96 dx 24 + 16.02 Then the integrand has partial fractions decomposition b CC +d + + 22 + 16 where a - 2 a = b = C = du Integrating term by term, we obtain that 16x3 + 10x2 + 64x + 96 dc 24 + 16x2 +C
To do this, we first need to factor the denominator of the integrand into linear factors. In this case, the denominator is given as 24 + 16.02 = 40, which is already a factorization. Therefore, we can write:
∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = ∫ [(a/(24 + 16.02)) + (b/(22 + 16))] dx
where a, b are constants that we need to find. To do this, we can use the method of partial fractions, which involves equating the coefficients of like terms on both sides of the equation. Specifically, we can write:
6x3 + 10x2 + 64x + 96 = (a/(24 + 16.02))(22 + 16) + (b/(22 + 16))(24 + 16.02)
Multiplying both sides by the common denominator (24 + 16.02)(22 + 16), we get:
(6x3 + 10x2 + 64x + 96)(24 + 16.02)(22 + 16) = a(22 + 16) + b(24 + 16.02)(24 + 16)
Expanding both sides and collecting like terms, we get a system of two linear equations in two unknowns:
(24 + 16.02)(22 + 16)a + (24 + 16.02)(24 + 16)b = 6(24 + 16.02)(22 + 16) + 10(22 + 16)(24 + 16.02) + 64(24 + 16.02) + 96(22 + 16)
(22 + 16)a + (24 + 16.02)b = 6(22 + 16) + 10(24 + 16.02) + 64 + 96
Solving this system (which involves some algebraic manipulation) gives:
a = -6/5
b = 18/5
Therefore, we can write:
∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = (-6/5) ∫ (22 + 16)/(24 + 16.02) dx + (18/5) ∫ (24 + 16.02)/(22 + 16) dx
To evaluate these integrals, we can use the substitution u = 24 + 16.02 in the first integral and u = 22 + 16 in the second integral. This gives:
∫ (6x3 + 10x2 + 64x + 96)/(24 + 16.02) dx = (-6/5) ln|24 + 16.02| + (18/5) ln|22 + 16| + C
where C is the constant of integration. Finally, using the given expression for the integral, we can equate coefficients of like terms to obtain:
16x3 + 10x2 + 64x + 96 = (6/5)(24 + 16.02) ln|24 + 16.02| - (18/5)(22 + 16) ln|22 + 16| + C
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Evaluate the integrals (Indefinite and Definite) and Simplify. 5 (a) 5 (5:-* - - 5 sin ) : dc xl1 (v) [(1822–1 18x)(6x3 – 9x2 – 3)6 dx ° ? (c) | Viana sec2 х dx (d) os Venta de Зх dx Væ+4 2 (e) ( 120 dax V1 + 2x2
(a) Indefinite integral of 5(5x^4 - 5sinx)dx is (5/3)x^5 + 5cosx + C. Definite integral over [0, π/2] is (125π/6) - 5.
We can evaluate the indefinite integral by applying the power rule and integration by substitution. The definite integral can be evaluated by substituting the limits of integration and simplifying.
(b) Indefinite integral of [(18x^2 - 1)(6x^3 - 9x^2 - 3)]^6dx is (18x^11 - 77x^9 + 126x^7 - 108x^5 + 49x^3 - 9x) / 11 + C.
To simplify the given expression, we can first expand the polynomial and then apply the power rule to integrate each term. The constant of integration can be added at the end.
(c) Definite integral of ∫tan^2(x)sec^2(x)dx over [0,π/4] is 1.
We can use the trigonometric identity sec^2(x) - 1 = tan^2(x) to simplify the integrand. Then we can apply the power rule and substitute the limits of integration to evaluate the definite integral.
(d) Indefinite integral of ∫(x+4)^2√(3x^2+4)dx is (1/15)(3x^2+4)^(3/2)(x+4) - (4/45)(3x^2+4)^(3/2) + C.
We can use substitution to simplify the integrand by setting u = 3x^2 + 4. After integrating, we can substitute back for u and simplify the constant of integration.
(e) Indefinite integral of ∫(120/(1+2x^2))dx is 60√2tan^(-1)(√2x) + C.
We can use substitution to simplify the integrand by setting u = 1 + 2x^2. After integrating, we can substitute back for u and simplify the constant of integration.
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A lake currently has a depth of 30 meters. As sediment builds up in the lake, its depth decreases by 2% per year.
This situation represents:
A. Exponential decay
B. Exponential growth
The rate of growth or decay, r, is equal to:
A. 1. 02
B. 0. 02
C. 0. 98
So the depth of the lake each year is ______ times the depth in the previous year.
A. 0. 98
B. 0. 02
C. 1. 02
It will take between _____ years for the depth of the lake to reach 26. 7 meters.
A. 3 and 4
B. 11 and 12
C. 9 and 10
D. 5 and 6
The situation represents exponential decay.
The rate of growth or decay, r, is equal to 0.02.
So the depth of the lake each year is 0.98 times the depth in the previous year.
It will take between 11 and 12 years for the depth of the lake to reach 26. 7 meters.
The situation represents exponential decay, as the depth of the lake decreases by a constant percentage each year. The rate of decay is 2% per year, so the rate of growth or decay, r, is equal to 0.98 (1 - 0.02). This means that the depth of the lake each year is 0.98 times the depth in the previous year.
To find the number of years it will take for the depth of the lake to reach 26.7 meters, we can use the formula for exponential decay:\
D = D₀ *[tex]e^{(-rt)[/tex]
where D is the current depth, D₀ is the initial depth, r is the rate of decay, and t is the number of years.
Substituting the given values, we get:
26.7 = 30 * [tex]e^{(-0.02t)[/tex]
Solving for t, we get:
t = ln(26.7/30) / (-0.02) ≈ 11.33
Therefore, it will take between 11 and 12 years for the depth of the lake to reach 26.7 meters.
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The length of a rectangle is 3 cm less than twice it’s width. the perimeter of the rectangle is 48cm
The length of the rectangle is 15 cm and the width is 9 cm.
What is the width of the rectangle?Let's start by setting up the equations we need to solve:
L = 2W - 3 (the length is 3 cm less than twice the width)
2L + 2W = 48 (the perimeter is 2 times the length plus 2 times the width)
Now we can substitute the first equation into the second equation and solve for W:
2(2W - 3) + 2W = 48
4W - 6 + 2W = 48
6W = 54
W = 9
Now that we know the width is 9 cm, we can substitute this value back into the first equation and solve for L:
L = 2(9) - 3
L = 15
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One side of an isosceles triangle is 2x + 1ft long. The other two sides are both 3x-14 long. The perimeter of the triangle is 55 ft. What is the length of each side? Show your work.
Let's use "a" to represent the length of the equal sides of the isosceles triangle, and let's use "b" to represent the length of the third side. We're told that one of the equal sides is 2x + 1ft long, so we can set up an equation:
2a + b = 55
We're also told that the other two sides are both 3x - 14ft long, so we can set up another equation:
a = 3x - 14
Now, we can substitute the second equation into the first equation and solve for "b":
2a + b = 55
2(3x-14) + b = 55
6x - 28 + b = 55
b = 83 - 6x
Now, we can substitute both equations into the equation a = 3x - 14 and solve for "x":
3x - 14 = 2x + 1 + 3x - 14
6x - 27 = 0
x = 4.5
Finally, we can substitute "x" into our equations to find the lengths of the sides:
a = 3x - 14 = 3(4.5) - 14 = 0.5
b = 83 - 6x = 83 - 6(4.5) = 55
So the length of the equal sides is 0.5ft, and the length of the third side is 55ft. Therefore, the lengths of the sides of the isosceles triangle are 0.5ft, 0.5ft, and 55ft.