For the given function y = 126 \sqrt(x), dy is 3.57 if x = 9 and Δx = dx = 0.17.
To find the change in y (or dy), we need to use the formula:
dy = f'(x) * dx
where f'(x) is the derivative of y with respect to x.
The given function is y = 126 \sqrt(x).
To find the derivative, we can use the power rule and chain rule of differentiation.
y = 126x^{1/2}
Taking the derivative of y with respect to x:
dy/dx = 1/2 * 126x^(-1/2)
Simplifying, we get:
dy/dx = 63/(\sqrt(x))
Now, we can substitute x = 9 into this expression to get the value of the derivative at that point:
dy/dx = 63/(\sqrt(9)) = 63/3 = 21
Next, we can use the given value of dx, which is 0.17, to find the change in y or dy.
dy = f'(x) * dx ≈ 21 * 0.17 ≈ 3.57
Therefore, the change in y or dy is approximately 3.57.
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Julia works at a music store. One of her jobs is to stock new CDs on the shelf. A recent order arrived with 215 classical CDs, 125 jazz CDs, and 330 soft rock CDs. How many groups will Julia use to arrange all of the CDs?
Julia will use 10 groups to arrange all of the CDs.
To determine the number of groups Julia will use to arrange all of the CDs, we need to find the greatest common divisor of the numbers 215, 125, and 330.
First, we can check if any of the numbers are divisible by 5:
215 is not divisible by 5
125 is divisible by 5 (125 ÷ 5 = 25)
330 is divisible by 5 (330 ÷ 5 = 66)
Now we divide 125 and 330 by 5:
125 ÷ 5 = 25
330 ÷ 5 = 66
Next, we check if any of the numbers are divisible by 2:
25 is not divisible by 2
66 is divisible by 2 (66 ÷ 2 = 33)
Now we divide 66 by 2:
66 ÷ 2 = 33
Therefore, the greatest common divisor of 215, 125, and 330 is 5 × 2 = 10.
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A square pyramid has a base that is 4 inches wide and a slant height of 7 inches. What is the surface area, in square inches, of the pyramid?
The surface area is 72 square inches.
To find the surface area of a square pyramid, we need to calculate the area of the base and the four triangular faces.
Given that the base is 4 inches wide, the area of the square base is:
Base area = side² = 4² = 16 square inches.
The slant height is 7 inches. To find the area of one triangular face, we use:
Triangle area = (base * slant height) / 2
Each triangle has the same base length as the square base, which is 4 inches. Therefore, the area of one triangular face is:
Triangle area = (4 * 7) / 2 = 14 square inches.
Since there are four triangular faces, their total area is:
4 * Triangle area = 4 * 14 = 56 square inches.
Finally, add the base area and the total area of the triangular faces to get the surface area:
Surface area = Base area + Total triangular faces area = 16 + 56 = 72 square inches.
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To find the quotient of 4. 082 and 10,000, move the decimal point in 4. 082
Choose.
places to the
Choose.
The Quotient of 4.082 and 10,000 is 0.0004082.
Find the qoutient of 4. 082 and 10,000?
To find the quotient of 4.082 and 10,000, we need to move the decimal point in 4.082 four places to the left, since there are four zeros in 10,000.
So, we get:
4.082 ÷ 10,000 = 0.0004082 is the answer.
Explanation.
To find the quotient of 4.082 and 10,000, we need to divide 4.082 by 10,000. When we divide by a number that is a power of 10, we can simplify the calculation by moving the decimal point to the left as many places as there are zeros in the divisor.
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Which equation models this relationship?
An equation that models this relationship include the following: C. t = 5d.
What is a proportional relationship?In Mathematics, a proportional relationship produces equivalent ratios and it can be modeled or represented by the following mathematical equation:
y = kx
Where:
k is the constant of proportionality.y represent the distance.x represent the time.Next, we would determine the constant of proportionality (k) for the data points contained in the table as follows:
Constant of proportionality, k = y/x = t/d
Constant of proportionality, k = 5/1
Constant of proportionality, k = 5.
Therefore, the required equation is given by;
t = kd
t = 5d
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
what dose y equal when the equation is negitive 5 y plus 4 is equal to negitive 11
Answer: y = 1.4
Step-by-step explanation:
if you write the equation it would be
-5y - 4 = -11
so first you would subtract 4 from -4 and -11 to cancel out the four.
so your equation would look like this -5y= -7
so now u would divide -5 by both sides to canceled out the -5
your equation should end up looking like
y=1.4
An employee at the metropolitan museum of art surveyed a random sample of 150 visitors to the museum. Of those visitors, 45 people bought food at the cafeteria. Based on those results, how many people out of 1750 visitors to the museum would be expected to buy food for the cafeteria? No links
We can expect that approximately 525 people out of 1750 visitors to the museum would buy food at the cafeteria.
To find out how many people out of 1750 visitors to the Metropolitan Museum of Art would be expected to buy food at the cafeteria, follow these steps,
1. Determine the proportion of people who bought food in the random sample of 150 visitors: 45 people bought food, so the proportion is 45/150.
2. Simplify the proportion: 45/150 = 0.3 or 30%.
3. Apply this proportion to the total number of 1750 visitors: 1750 * 0.3 = 525.
So, based on the survey results, we can expect that approximately 525 people out of 1750 visitors to the museum would buy food at the cafeteria.
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In survey 55%of those surveyed said that they get news from local television station,three-fifths said that they get the news from daily news paper and 0. 4 said they get they get their news form the internet. Which new source has the most users
The daily newspaper has the most users among those surveyed.
To determine which news source has the most users, we need to compare the percentages of those who use each source.
According to the survey:
55% get news from local television station
60% get news from daily newspaper
40% get news from the internet
To compare these percentages, we can either convert them to fractions or decimals. Let's convert them to decimals:
55% = 0.55
60% = 0.60
40% = 0.40
Now we can compare them directly. We see that the source with the highest percentage is the daily newspaper, with 60% of those surveyed saying they get news from it. Therefore, the daily newspaper has the most users among the surveyed population.
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Please Help Quick!!!!
On a coordinate plane, a line segment has endpoints P(6,2) and Q(3. 8). 9. Point M lies on PQ and divides the segment so that the ratio of PM-MQ is 2-3. What are the coordinates of point M?
The coordinates of point M come out to be 4.8, 4.4
This case is solved by using the section formula which states that
The coordinate of point P that divides the line segment AB in the ratio of m:n where the coordinate of A is [tex]x_1,y_1[/tex] and the B is [tex]x_2,y_2[/tex] is described as
[tex]\frac{mx_2+nx_1}{m+n}[/tex],[tex]\frac{my_2+ny_1}{m+n}[/tex]
The line to be divided = PQ
Coordinates of P = (6,2)
Coordinates of Q = (3,8)
Ratio = 2:3
Thus, the coordinates of M = [tex]\frac{2*3+3*6}{2+3}[/tex],[tex]\frac{8*2+2*3}{2+3}[/tex]
= 24/5 , 22/5
= 4.8, 4.4
Point M with coordinates (4.8,4.4) lies on PQ and divides the segment so that the ratio of PM-MQ is 2-3
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In one month 382 adults and 65 children stayed in a hotel. How many people are there altogether?
In one month, a total of 447 people stayed at the hotel.
In one month, a hotel had 382 adults and 65 children staying as guests.
To find out the total number of people who stayed at the hotel, we simply need to add the number of adults and children together.
In one month, a total of 447 people (382 adults and 65 children) stayed at the hotel.
Overall, this problem is a simple example of addition in action. By adding the number of adults and children together, we can determine the total number of people who stayed in the hotel.
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Let a = (- 2, 4, 2) and b = (1, 0, 3).
Find the component of b onto a
The component of b onto a is (-1/3, 2/3, -1/3).
To find the component of b onto a, we first need to find the projection of b onto a. The projection of b onto a is given by the formula:
proj_a(b) = (b dot a / ||a||^2) * a
where dot represents the dot product and ||a|| represents the magnitude of vector a.
We can calculate the dot product of a and b as follows:
a dot b = (-2*1) + (4*0) + (2*3) = 4
We can calculate the magnitude of a as follows:
||a|| = sqrt((-2)^2 + 4^2 + 2^2) = sqrt(24) = 2sqrt(6)
Now we can plug these values into the formula for the projection of b onto a:
proj_a(b) = (b dot a / ||a||^2) * a
proj_a(b) = (4 / (2sqrt(6))^2) * (-2, 4, 2)
proj_a(b) = (4 / 24) * (-2, 4, 2)
proj_a(b) = (-1/3, 2/3, -1/3)
Finally, the component of b onto a is simply the projection of b onto a:
comp_a(b) = (-1/3, 2/3, -1/3)
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The Willis tower in Chicago is the second tallest building in the United States in his topped by a high intent. A surveyor on the ground makes the following measurements. The angle of elevation from her position to the top of the building is 34°. The distance from her position to the top of the building is 2595 feet. The distance from her position to the top of the antenna is 2760 feet. how far away from the base of the building is the surveyor located? How tall is the building? What is the angle of elevation from the surveyor to the top of the antenna? How tall is the antenna?
The surveyor is located about 239.6 feet away from the base of the Willis Tower.
The height of the Willis Tower is 165 feet.
The angle of elevation from the surveyor to the top of the antenna is about 3.41°.
The height of the antenna is about 135.9 feet.
How to solve for the angle of elevationLet's call the distance from the surveyor to the base of the Willis Tower "x", and let's call the height of the Willis Tower "h".
We can use trigonometry to solve for x and h. First, let's find x:
tan(34°) = h/x
x = h/tan(34°)
Now we can use the distance from the surveyor to the top of the building to solve for h:
h + 2595 = 2760
h = 165
So the height of the Willis Tower is 165 feet. Now we can solve for x:
x = 165/tan(34°) ≈ 239.6 feet
So the surveyor is located about 239.6 feet away from the base of the Willis Tower.
To find the angle of elevation from the surveyor to the top of the antenna, we can use trigonometry again:
tan(θ) = h/2760
θ = tan^(-1)(h/2760)
θ ≈ 3.41°
So the angle of elevation from the surveyor to the top of the antenna is about 3.41°.
Finally, we can use the height of the Willis Tower and the distance from the surveyor to the top of the antenna to solve for the height of the antenna:
tan(34°) = (h + a)/2760
a ≈ 135.9
So the height of the antenna is about 135.9 feet.
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We want to conduct a hypothesis test of the claim that the population mean time it takes drivers to react following the application of brakes by the driver in front of them is more than 2. 5 seconds. So, we choose a random sample of reaction time measurements. The sample has a mean of 2. 4 seconds and a standard deviation of 0. 5 seconds. For each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean. Then calculate that statistic. Round your answers to two decimal places. (a) The sample has size 110, and it is from a non-normally distributed population with a known standard deviation of. It is unclear which test statistic to use. (b) The sample has size 12,and it is from a normally distributed population with an unknown standard deviation. Z=
t=
It is unclear which test statistic to use.
(a) We will use the "t" as test-statistics and the value of "t" is t = -0.87.
(b) We will use "z" as the test-statistics and the value of "z" is z = -0.77.
In statistics, a test statistic is a numerical summary of a sample that is used to make an inference about a population parameter. It is calculated from the sample data and is used to test a hypothesis or to make a decision about some characteristic of the population.
Part (a) : In this case, we do not know the "standard-deviation",
the case in which standard-deviation is un-known, "t" is used as a "test-statistics.
The Standard-error-of-mean (SE) is = s/√n = 0.5/√19 = 0.1148.
So, "t" = (mean - 2.5)/SE,
Substituting the values,
We get,
t = (2.4 - 2.5)/0.1148 = -0.87.
Part (b) : In this case, we know the "standard-deviation",
The case in which standard-deviation is known, "z" is used as a "test-statistics.
The Standard-error-of-mean (SE) is = s/√n = 0.45/√12 = 0.1299.
So, "z" = (mean - 2.5)/SE,
Substituting the values,
We get,
z = (2.4 - 2.5)/0.1299 = -0.77.
Therefore, the value of "z" is -0.77.
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The given question is incomplete, the complete question is
We want to conduct a hypothesis test of the claim that the population mean time it takes drivers to react following the application of brakes by the driver in front of them is more than 2.5 seconds. So, we choose a random sample of reaction time measurements. The sample has a mean of 2.4 seconds and a standard deviation of 0.5 seconds.
For each of the following sampling scenarios, choose an appropriate test statistic for our hypothesis test on the population mean. Then calculate that statistic. Round your answers to two decimal places.
(a) The sample has size 19, and it is from a non-normally distributed population with an unknown standard deviation.
Which test-statistic will you use z, t or It is unclear which test statistic to use.
(b) The sample has size 12,and it is from a normally distributed population with an known standard deviation of 0.45.
Which test-statistic will you use z, t or It is unclear which test statistic to use.
the variables x and y vary inversely. use the given values to write an equation relating i and y. then find y when i = i= 5, y = -4 an equation is y= when i = 3, y = 5
please help me!
When i (x) = 3, the value of y is approximately -6.67. The equation relating i (x) and y in this inverse variation is xy = -20.
The given information states that the variables x and y vary inversely. To write an equation relating i (assuming it's x) and y, we first need to understand the concept of inverse variation.
In inverse variation, the product of the two variables remains constant. Mathematically, it can be represented as xy = k, where k is the constant of variation. We are given the values i (x) = 5 and y = -4. Using these values, we can find the constant of variation, k:
5 * -4 = k
k = -20
Now that we have the constant of variation, we can write the equation relating i (x) and y as:
xy = -20
Next, we want to find the value of y when i (x) = 3. We can use the equation we just derived to find the value of y:
3 * y = -20
Now, we can solve for y:
y = -20 / 3
y ≈ -6.67
So, when i (x) = 3, the value of y is approximately -6.67. The equation relating i (x) and y in this inverse variation is xy = -20.
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Y=1/3x-3 and y=-x+1 what the answer pls i really need this
The point of intersection between the two given equations is (3, -2).
The problem is asking to find the point of intersection between the two given equations:
y = (1/3)x - 3 ............... (equation 1)
y = -x + 1 ............... (equation 2)
To solve for the intersection point, we can set the two equations equal to each other:
(1/3)x - 3 = -x + 1
Simplifying and solving for x:
(1/3)x + x = 1 + 3
(4/3)x = 4
x = 3
Now that we know x = 3, we can substitute it into either of the two original equations to find y:
Using equation 1: y = (1/3)x - 3 = (1/3)(3) - 3 = -2
Using equation 2: y = -x + 1 = -(3) + 1 = -2
Therefore, the intersection point is (3, -2).
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Bill needs a table to display his model train set. the table needs to be 2 times longer and 3 inches shorter
than it is wide and have an area of 4,608 square inches. what does x need to be to fit these requirements?
2x-3
2x - 3 would be 92 - 3 = 89 inches, which is the length of the table
How to find the length?.The table needs to be 2 times longer than it is wide, so its length is 2 times its width, or 2x.
The table also needs to be 3 inches shorter than it is wide, so its width is x + 3 inches.
The area of the table is 4,608 square inches, so we can set up an equation:
2x(x + 3) = 4,608
Simplifying this equation:
2x²+ 6x = 4,608
Dividing both sides by 2:
x²+ 3x - 2,304 = 0
We can solve for x using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In this case, a = 1, b = 3, and c = -2,304. Substituting these values:
x = (-3 ± √(3² - 4(1)(-2,304))) / 2(1)
Simplifying:
x = (-3 ± √(9 + 9,216)) / 2
x = (-3 ± √(9,225)) / 2
x = (-3 ± 95) / 2
x = 46 or x = -49
Since the width of the table cannot be negative, we can ignore the negative solution. Therefore, x needs to be 46 inches to fit the given requirements.
The length of the table is 2x, or 2(46) = 92 inches, and the width is x + 3, or 46 + 3 = 49 inches. The area is 92 * 49 = 4,508 square inches, which matches the given area requirement.
So, 2x - 3 would be 92 - 3 = 89 inches, which is the length of the table.
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Janice bought a new car. the total amount she needs to borrow is $35,000 . she plans on taking out a 5-year loan at an apr of 4%. what is the monthly payment ?
Janice's monthly payment for her 5-year, 4% APR car loan would be $626.38.
To calculate Janice's monthly payment, we first need to use the formula for calculating loan payments:
Loan Payment = Loan Amount / Discount Factor
The discount factor can be calculated using the following formula:
Discount Factor = [(1 + r)ⁿ] - 1 / [r(1 + r)ⁿ]
Where r is the monthly interest rate (4% divided by 12 months = 0.00333) and n is the total number of payments (5 years x 12 months = 60).
Plugging in the values, we get:
Discount Factor = [(1 + 0.00333)⁶⁰] - 1 / [0.00333(1 + 0.00333)⁶⁰] = 55.8389
Now, we can calculate Janice's monthly payment:
Loan Payment = $35,000 / 55.8389 = $626.38
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If you were to randomly survey 20 people at 50 random high schools would this be a random sample? Why or why not?
Yes, if you were to randomly survey 20 people at 50 random high schools, it would be considered a random sample because the process involves randomly selecting people from randomly selected high schools, which prevents selection bias..
A random sample is a subset of a population in which every individual has an equal chance of being selected. In this case, the population is the students at the high schools.
By randomly selecting the 50 high schools, you ensure that each school has an equal opportunity to be part of the sample. This helps to prevent selection bias, as no specific schools are deliberately chosen. Moreover, by surveying 20 random people within each selected school, you further eliminate bias, as each student at the school has an equal chance of being selected for the survey.
This random sampling method is beneficial because it helps to obtain a more representative sample of the larger population of high school students. By including diverse schools and students, the survey results can provide more accurate and generalizable insights.
However, it is important to note that even with random sampling, there may still be some limitations, such as sampling error or non-response bias. To minimize these, it is essential to ensure that the sample size is large enough and that survey procedures are properly executed.
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Sand falls from an overhead bin and accumulates in a conical pile with a radius that is alwavs two times its heiaht. Suppose the height of the pile increases at a rate of 2 cm/s when the pile is 11 cm high. At what rate is the sand leaving the bin at that instant?
To solve this problem, we need to use related rates. Let's start by drawing a diagram:
```
/\
/ \
/ \
/ \
/ \
/__________\
```
We know that the radius of the conical pile is always two times its height, so we can label the diagram as follows:
```
/\
/ \
/ \
/ \
/ \
/__________\
/| r=2h \
/ |___________\
```
Now we need to find an equation that relates the height of the pile to its radius. We can use the formula for the volume of a cone:
```
V = (1/3)πr^2h
```
We want to solve for h in terms of r:
```
V = (1/3)πr^2h
3V/πr^2 = h
```
Now we can differentiate both sides of this equation with respect to time:
```
d/dt (3V/πr^2) = d/dt h
0 = (3/πr^2) dV/dt - (2/πr^3) dr/dt
```
We're given that the height is increasing at a rate of 2 cm/s when the pile is 11 cm high, so we know that:
```
dh/dt = 2 cm/s
h = 11 cm
```
We want to find the rate at which sand is leaving the bin, which is given by `dV/dt`. We can solve for this using the equation we derived:
```
0 = (3/πr^2) dV/dt - (2/πr^3) dr/dt
dV/dt = (2/3)πr^2 (dh/dt) / r
```
Now we just need to plug in the values we know:
```
dh/dt = 2 cm/s
h = 11 cm
r = 2h = 22 cm
dV/dt = (2/3)π(22)^2 (2) / 22
dV/dt = 264π/3
```
So the rate at which sand is leaving the bin when the pile is 11 cm high is `264π/3 cm^3/s`.
To solve this problem, we can use the relationship between the radius and height of the conical pile, as well as the given rate of height increase.
Since the radius (r) is always two times the height (h), we have r = 2h. The volume (V) of a cone is given by the formula V = (1/3)πr^2h. We can substitute r with 2h, so V = (1/3)π(2h)^2h.
Now, let's differentiate both sides with respect to time (t):
dV/dt = (1/3)π(8h^2)dh/dt
When the height is 11 cm, the rate of height increase (dh/dt) is 2 cm/s. We can substitute these values into the equation:
dV/dt = (1/3)π(8(11)^2)(2)
Solving for dV/dt:
dV/dt ≈ 2046.92 cm³/s
At that instant, the sand is leaving the bin at a rate of approximately 2046.92 cm³/s.
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Find an equation of the circle drawn below.
The equation of the circle in this problem is given as follows:
x² + y² = 49.
What is the equation of a circle?The equation of a circle of center [tex](x_0, y_0)[/tex] and radius r is given by:
[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]
The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle. As this distance is of 7 units, it is then given as follows:
r = 7 -> r² = 49.
The center of the circle is at the origin, hence:
[tex](x_0, y_0) = (0,0)[/tex]
Thus the equation of the circle is given as follows:
x² + y² = 49.
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If a woman making $29,000 a year receives a cost-of-living increase of 2. 6%, what will her new salary be?
To find the new salary after a 2.6% increase, we need to add 2.6% of the original salary to the original salary.
2.6% of $29,000 can be calculated as:
(2.6/100) x $29,000 = $754
Therefore, the new salary will be:
$29,000 + $754 = $29,754
So the woman's new salary will be $29,754.
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Find the area of the shaded region. Round to final answer to the nearest tenth for this problem.
Answer:
(1/6)π(4^2) - (1/2)(2√3)(4)
= 8π/3 - 4√3 = about 1.4
You can use indirect measurement to estimate the height of a building. First, measure your distance from the base of the building and the distance from the ground to a point on the building that you are looking at. Maintaining the same angle of sight, move back until the top of the building is in your line of sight. Answer both A and B
The building is perfectly vertical and the observer is at a consistent height above the ground.
A) Explain how the method of indirect measurement can be used to estimate the height of a building?The method of indirect measurement can be used to estimate the height of a building by using similar triangles and the principles of proportionality. First, the distance from the base of the building to the observer and the distance from the ground to a known point on the building are measured. By maintaining the same angle of sight, the observer can move back until the top of the building is in their line of sight. At this point, a second pair of measurements is taken: the distance from the new location to the base of the building and the height of the visible portion of the building from the ground. By using the principles of proportionality between similar triangles, the height of the entire building can be estimated.
Specifically, the ratio of the height of the known point on the building to the distance from the observer to that point can be set equal to the ratio of the height of the entire building to the distance from the observer to the base of the building. This proportion can be solved algebraically to find the estimated height of the entire building.
B) What are some potential sources of error or inaccuracy in this method of estimation?
There are several potential sources of error or inaccuracy in this method of estimation. One major source of error is the assumption that the two triangles being compared are similar. If the angle of sight is not maintained exactly or if the ground is not perfectly level, the triangles may not be similar and the estimated height may be incorrect.
Additionally, the accuracy of the estimated height depends on the accuracy of the distance measurements. If the distances are not measured precisely, the estimated height will be proportionally less accurate.
Finally, this method assumes that the building is perfectly vertical and that the observer is at a consistent height above the ground. If the building is not perfectly vertical or the observer's height above the ground changes, this can also affect the accuracy of the estimated height.
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How many different simple random samples of size 4 can be obtained from a population whose size is 50?
The number of random samples, obtained using the formula for combination are 230,300 random samples
What is a random sample?A random sample is a subset of the population, such that each member of the subset have the same chance of being selected.
The formula for combinations indicates that we get;
nCr = n!/(r!*(n - r)!), where;
n = The size of the population
r = The sample size
The number of different simple random samples of size 4 that can be obtained from a population of size 50 therefore can be obtained using the above equation by plugging in r = 4, and n = 50, therefore, we get;
nCr = 50!/(4!*(50 - 4)!) = 230300
The number of different ways and therefore, the number of random samples of size 4 that can be selected from a population of 50 therefore is 230,300 random samples.
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Can someone please help me ASAP? It’s due tomorrow. I will give brainliest if it’s correct.
The probability values when calculated are
P(2 numbers greater than 3) = 0.1P(2 even numbers) = 0.4P(2 cards with same numbers) = 0P(1 card is 3) = 0.3Evaluating the probability valuesFrom the question, we have the following parameters that can be used in our computation:
Cards = {1, 2, 3, 4, 5}
Selecting two cards without replacement
So, we have
P(2 numbers greater than 3) = 2/5 * 1/4
P(2 numbers greater than 3) = 0.1
P(2 even numbers) = 2/5 * 4/4
P(2 even numbers) = 0.4
P(2 cards with same numbers) = 1/5 * 0/4
P(2 cards with same numbers) = 0
P(1 card is 3) = 2 * 1/5 * 3/4
P(1 card is 3) = 0.3
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Solve the equation 2^(x-2)+2^3-x=3. Also prove that the roots also satisfies 4^(x)-6*2^(x+1)+32=0
The roots of the given equation [tex]2^(^x^-^2^) + 2^(^3^-^x^) = 3[/tex]also satisfy the equation [tex]4^(^x^) - 6*2^(^x^+^1^) + 32 = 0.[/tex]
How to find the roots of equation?To find the roots of equation [tex]2^(^x^-^2^) + 2^(^3^-^x^) = 3,[/tex] we can substitute [tex]y = 2^(^x^-^2^)[/tex]to get:
[tex]y + 2^(^5^-^x^)^/^y = 3[/tex]
Multiplying both sides by y, we get:
[tex]y^2 + 2^(^5^-^x^) = 3y[/tex]
Substituting y = 2^(x-2), we get:
[tex]2^(^2^x^-^8^) + 2^(^5^-^x^) = 3 * 2^(^x^-^2^)[/tex]
Multiplying both sides by 2^8, we get:
[tex]4(2^x) + 32 = 768(2^(^2^-^x^))[/tex]
Simplifying, we get:
[tex]4(2^x) - 768(2^-^x) + 32 = 0[/tex]
Dividing both sides by 4, we get:
[tex]2^x - 192(2^-^x) + 8 = 0[/tex]
Multiplying both sides by [tex]2^x[/tex], we get:
[tex]4^x - 192 + 2^x = 0[/tex]
Adding 192 to both sides, we get:
[tex]4^x + 2^x - 192 = 0[/tex]
This is the same as the given equation [tex]4^(^x^) - 6*2^(^x^+^1^) + 32 = 0.[/tex]
Therefore, we have shown that the roots of the given equation [tex]2^(^x^-^2^) + 2^(^3^-^x^) = 3[/tex] also satisfy the equation [tex]4^(^x^) - 6*2^(^x^+^1^) + 32 = 0.[/tex]
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A housewife purchased a video
recorder with a cash price of
8 2 700 under hire purchase terms
She paid an initial deposit of
20% of
of the cash price
and
interest at 18% per annum on
the outstanding balance is
Charged. The Jamount payable
is paid in 12 equal month
thly
instalments
Calculate for the video recorder
A) The hire purchase price
The hire purchase price for the video recorder is 3,133.20.
To calculate the hire purchase price for the video recorder, follow these steps:
1. Calculate the initial deposit: 20% of the cash price (2,700) is (0.20 * 2,700) = 540.
2. Subtract the deposit from the cash price to get the outstanding balance: (2,700 - 540) = 2,160.
3. Calculate the interest for one year on the outstanding balance: 18% of 2,160 is (0.18 * 2,160) = 388.80.
4. Divide the interest by 12 to find the interest per month: (388.80 / 12) = 32.40.
5. Add the interest per month to the outstanding balance: (2,160 + 32.40 * 12) = 3,133.20.
6. The hire purchase price is 3,133.20, which is the total amount payable in 12 equal monthly instalments.
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Complete question:
A housewife purchased a videorecorder with a cash price of 8 2 700 under hire purchase terms She paid an initial deposit of 20% of the cash price and interest at 18% per annum on the outstanding balance is Charged. The Jamount payable is paid in 12 equal monththly instalments Calculate for the video recorder The hire purchase price
Val needs to find the area enclosed by the figure. The figure is made by attaching semicircles to each side of a 56m-by-56m square. Val says the area is 1,787.52m. Find the area enclosed by the figure. Use 3.14 for . What error might have made?
Val's calculation of 1,787.52 m² is incorrect.
What is area of semicircle?
The area of a semicircle is half the area of the corresponding circle. If r is the radius of the semicircle, then the area of the semicircle is:
A(semicircle) = (1/2) π r²
To find the area enclosed by the figure, we need to add the areas of the square and the four semicircles.
The area of the square is:
[tex]A_{square}[/tex] = (56 m)² = 3,136 m²
The area of one semicircle is half the area of the corresponding circle, and the radius of the circle is equal to the side length of the square. Therefore, the area of one semicircle is:
[tex]A_{semicircle}[/tex] = (1/2) π (56/2)²= 1,554.56 m²
The total area enclosed by the figure is:
[tex]A_{total}[/tex] = [tex]A_{square}[/tex]+ 4 [tex]A_{semicircle}[/tex] = 3,136 + 4(1,554.56) = 9,901.44 m²
Therefore, Val's calculation of 1,787.52 m² is incorrect.
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Question:
Val needs to find the area enclosed by the figure. The figure is made by attaching semicircles to each side of a 56m-by-56m square. Val says the area is 1,787.52m. Find the area enclosed by the figure. Use 3.14 for π. What error might have Val made?
The vector v and its initial point are given. Find the terminal point.
v = (3, -6, 6)
Initial point: (0, 6, 1)
(x,y,z) = ______
The terminal point (x, y, z) of vector v with the given initial point is (3, 0, 7).
To find the terminal point of vector v with initial point given, you can follow these steps:
Add the vector components to the coordinates of the initial point.
The vector v is given as (3, -6, 6) and the initial point is (0, 6, 1).
Add the x-components: 0 + 3 = 3
Add the y-components: 6 + (-6) = 0
Add the z-components: 1 + 6 = 7
The terminal point (x, y, z) of vector v with the given initial point is (3, 0, 7).
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The number of fish in a lake is growing exponentially. The table shows the values, in thousands, after different numbers of years since the population was first measured.
years population
0 10
1
2 40
3
4
5
6
By what factor does the population grow every two years? Use this information to fill out the table for 4 years and 6 years.
By what factor does the population grow every year? Explain how you know, and use this information to complete the table
The population in 4 year is 160 and in 6 year is 320.
The population is grows by a factor of 2 every two years.
We can use the following formula to get the rate of population growth every two years:
Growth factor: (population after n years / (population after (n-2) years) ^ 1/2
This formula can be used to determine the growth factor as follows:
Growth factor is equal to (40/10)*(1/2) = 2
This indicates that every two years, the population increases by a factor of 2.
Then, for 4 year the population
= 10 x 2⁴
= 10 x 16
= 160
For 6 year the population is
= 10 x 2⁶
= 10 x 32
= 320
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