The problem is solved by using calculus to find the maximum of the area function for a rectangle. The dimensions that yield maximum area with 24 yards of fencing are a length of 6 yards and a width of 12 yards, resulting in an area of 72 square yards.
Explanation:The problem involves maximizing the area of a rectangle with a fixed perimeter. Let's define the length of the rectangle as x and the width as y. Since you have 24 yards of fencing and you need to enclose three sides of the rectangle, the perimeter equation becomes 2x + y = 24, which can be rearranged as y = 24 - 2x.
The area of the rectangle can be represented by the equation A = xy, and substituting the y value from our perimeter equation, we get A = x(24 - 2x). To maximize this area, we take the derivative of A with respect to x and set it equal to zero, yielding the equation 24 - 4x = 0, or x = 6. Substituting x = 6 back into the y equation gives y = 12.
Therefore, the dimensions that maximize the area of the deck are a length of 6 yards and a width of 12 yards. Substituting these values back into the area equation gives a maximum area of 72 square yards.
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**HELP** ILL GIVE YOU 50 POINTS TO HELP ME
use the vertical line test to determine if the relation [(-6 -2),(-2 6),(0,3),(3,5)] is a function. explain your reason
Answer:
It is a function.
Step-by-step explanation:
[(-6 -2),(-2 6),(0,3),(3,5)]
Plot the points on a coordinate plain.
Draw vertical lines through the graph.
If any of the lines passes through the relation on more than one point it is not a function.
If not, it is a function.
Answer: It is a function
Step-by-step explanation: The vertical line test is used to determine if a relation is a function. If there are 2 points when you draw the line, then it is not a function, because a function is linear and cannot be vertical. You can solve this question is simply by looking at the numbers, and you can see that every input (x-value) has one output (y-value), and that there are no x-values that are repeated. So, therefore, it would be a function.
Many states have banned texting while driving because it is dangerous. Suppose you are driving 65 mph and you take your eyes off the road for four seconds. How many feet will you travel in that time. What is the dimensional analysis
65 mph means in 1 hour you drive 65 miles.
To find out how many feet you drive in 1 second you need to convert the above in terms of feet and secs
1 hour = 60 minutes and 1 minute has 60 secs
So 1 hour has 60* 60 secs = 3600 secs.
1 mile - 5280 feet
65 miles = 65*5280 feet
So in 3600 secs you are driving 65 * 5280 feet
When traveling at 65 mph and taking their eyes off the road for four seconds, the driver will cover approximately 4.33 miles in that time.
When driving at a speed of 65 mph, the driver is covering 65 miles in one hour. To find the distance traveled in four seconds, we need to convert the time from seconds to hours, as the speed is given in miles per hour.
Dimensional analysis:
Given speed: 65 miles per hour (65 mph)
Time taken: 4 seconds
Step 1: Convert seconds to hours
1 minute = 60 seconds
1 hour = 60 minutes
4 seconds = 4/60 minutes = 0.0667 hours
Step 2: Calculate the distance traveled
Distance = Speed × Time
Distance = 65 mph × 0.0667 hours
Detailed calculation:
Distance = 65 mph × 0.0667 hours
Distance ≈ 4.33455 miles
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Which function could this graph represent?
O y=2sin x+2
O y=-2sin x+ 2
O y= 2cos x + 2
O y=-2cos x + 2
Answer:
y = -2sin(x) + 2
Step-by-step explanation:
The sin(x) function is zero at x=0, so you want a sine function of some sort. The slope of your graph at x=0 is negative so you want a sine function with a negative multiplier. The appropriate choice is the one shown above.
__
The cos(x) function is 1 at x=0, so any graph involving the cosine will not go through the point (0, 2) the way your graph does.
Frank borrows 12000 dollars at 7.4 percent annual interest, compounded once per year, when he begins college. Four years later, how much will he owe? (Assume that he makes no payments during the four years.)
Answer: $15966.08
Step-by-step explanation:
The formula to calculate the compound amount is given by :-
[tex]A=P(1+r)^t[/tex]
, where P = initial deposit amount.
Time = Time period .
r= Rate of interest in decimal (compounded once per year)
Given : P= $ 12000
r= 7.4 percent =0.074
t= 4 years
Then, the compound amount after 4 years :
[tex]A=12000(1+0.074)^{4}\\\\=12000(1.074)^4=12000(1.33050688258)\\\\=15966.082591\approx15966.08[/tex]
Hence, compound amount after 4 years = $15966.08
After four years, Frank will owe approximately $16,140.
To determine the amount Frank will owe after four years, we can use the formula for compound interest:
[tex]\[ A = P(1 + \frac{r}{n})^{nt} \][/tex]
Given:
- P = $12,000 (the initial amount borrowed)
- r = 7.4% = 0.074 (annual interest rate)
- n = 1 (compounded once per year)
- t = 4 years (time period in question)
Plugging these values into the compound interest formula, we get:
[tex]\[ A = \$12,000(1 + \frac{0.074}{1})^{1 \times 4} \] \[ A = \$12,000(1 + 0.074)^{4} \] \[ A = \$12,000(1.074)^{4} \][/tex]
Now, we calculate [tex]\( (1.074)^{4} \)[/tex]:
[tex]\[ A = \$12,000 \times 1.345 \] \[ A \approx \$16,140 \][/tex]
Finally, we multiply this by the principal amount to find out how much will be owed after four years:
[tex]\[ A = \$12,000 \times 1.345 \] \[ A \approx \$16,140 \][/tex]
pls help me finna mark brainliest
The right answer is Option D.
Step-by-step explanation:
Given,
Total people surveyed = 250
Total people who prefer sports channel= 62
Percent of people who prefer sports channel;
Percent = [tex]\frac{people\ who\ prefer\ sports\ channel}{Total\ no.\ of\ people\ surveyed}*100[/tex]
[tex]Percent=\frac{62}{250}*100\\\\Percent=\frac{6200}{250}\\\\Percent= 24.8\%[/tex]
24.8% of everyone surveyed preferred sports channel.
The right answer is Option D.
Keywords: percentage, division
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A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.05 in.
(a) Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Assume the population standard deviation is 0.20 in.
(b) Repeat part (a) using a standard deviation of 0.40 in. Which standard deviation requires a larger sample size? Explain.
(a) The minimum sample size required to construct a 99% confidence interval using a population standard deviation of 0.20 in. is balls. (Round up to the nearest integer.)
(b) The minimum sample size required to construct a 99% confidence interval using a standard deviation of 0.40 in. is balls. (Round up to the nearest integer.)
A population standard deviation of in. requires a larger sample size. Due to the increased variability in the population, a sample size is needed to ensure the desired accuracy.
Answer:
107,426, bigger
Step-by-step explanation:
Given that a soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.05 in.
Margin of error = 0.05 inches
Since population std deviation is known we can use z critical value.
(a) i.e. for 99% confidence interval
Z critical = 2.58
[tex]2.58(\frac{0.20}{\sqrt{n} } )<0.05\\n>106.50\\n>107[/tex]
A minimum sample size of 107 needed.
b) [tex]2.58(\frac{0.40}{\sqrt{n} } )<0.05\\\\\\n>426[/tex]
Here minimum sample size = 426
Due to the increased variability in the population, a bigger sample size is needed to ensure the desired accuracy.
Final answer:
To determine the minimum sample size required for a 99% confidence interval for the mean circumference of soccer balls, use the formula n = (Z * σ / E) ^ 2. A standard deviation of 0.40 in requires a larger sample size than a standard deviation of 0.20 in.
Explanation:
To determine the minimum sample size required to construct a 99% confidence interval for the population mean, we can use the formula:
n = (Z * σ / E) ^ 2
Where:
n is the sample size
Z is the Z-score corresponding to the desired confidence level (in this case, 99% confidence level)
σ is the population standard deviation
E is the desired margin of error (0.05 in)
(a) The minimum sample size required to construct a 99% confidence interval using a population standard deviation of 0.20 in. is 82 balls.
(b) The minimum sample size required to construct a 99% confidence interval using a population standard deviation of 0.40 in. is 324 balls.
A standard deviation of 0.40 in requires a larger sample size. This is because a larger standard deviation indicates more variability in the population, which necessitates a larger sample size to ensure the desired level of accuracy.
PLZ HELP!!
The revolving restaurant on top of a hotel in San Francisco, California takes 45 minutes to complete a full counterclockwise rotation. A table that is 30 ft from the center of the restaurant starts at position (30, 0). What are the coordinates of the table after 9 minutes? Round to the nearest tenth.
A. (9.3, 28.5)
B. (28.5, 9.3)
C. (23, 19.3)
D. (11.3, 17.3)
Answer:
Step-by-step explanation:
In 9 minutes it would make 9/45 = 1/5 th of a revolution.
360(1/5) = 72 degrees
Coordinates:
(30cos72, +/- 30sin72) [+ for counterclockwise, - for clockwise)
(9.3ft, +/- 28.5ft)
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The coordinates of the table after 9 minutes are approximately (9.3, 28.5).
What are Coordinates?Coordinates are a collection of numbers that aid in displaying a point's precise location on the coordinate plane.
Since the restaurant takes 45 minutes to complete a full counter clockwise rotation, its angular velocity is:
ω = (2π radians) / (45 minutes)
≈ 0.1396 radians per minute
If we let θ be the angle between the position of the table and the positive x-axis at time t, then the position of the table can be expressed as:
x = 30 cos(θ)
y = 30 sin(θ)
To find the position of the table after 9 minutes, we can use the angular velocity to determine the angle that the restaurant has rotated. After 9 minutes, the angle of rotation is:
θ = ωt = 0.1396 radians/minute x 9 minutes
≈ 1.256 radians
Using the values of θ and the radius of 30 ft, we can find the coordinates of the table:
x = 30 cos(1.256) ≈ 9.3
y = 30 sin(1.256) ≈ 28.5
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The distribution of the annual incomes of a group of middle management employees approximated a normal distribution with a mean of $37,200 and a standard deviation of $800. About 68 percent of the incomes lie between what two incomes?
a. $30,000 and $40,000
b. $36,400 and $38,000
c. $34,800 and $39,600
d. $35,600 and $38,800
Answer:
Option B.
Step-by-step explanation:
Given information:
A group of middle management employees approximated a normal distribution.
Population mean [tex]\mu[/tex] = $37,200
Population standard deviation [tex]\sigma[/tex] = $800
About 68 percent of the incomes lie between two incomes and we need to find those two incomes.
We know that 68% data lies in the interval [tex][\mu-\sigma,\mu+\sigma][/tex].
[tex]\mu-\sigma=37,200-800=36,400[/tex]
[tex]\mu+\sigma=37,200+800=38,000[/tex]
About 68 percent of the incomes lie between what two incomes $36,400 and $38,000.
Therefore, the correct option is B.
Answer: b. $36,400 and $38,000
A Hospital/Surgical Expense policy was purchased for a family of four in March of 2013. The policy was issued with a $500 deductible and a limit of four deductibles per calendar year. Two claims were paid in September 2013, each incurring medical expenses in excess of the deductible. Two additional claims were filed in 2014, each in excess of the deductible amount as well. What would be this family's out-of-pocket medical expenses for 2013?
Answer:
The answer is $1000.
Step-by-step explanation:
The policy was issued with a $500 deductible and a limit of four deductibles per calendar year.
As given that two claims were paid in September 2013, each incurring medical expenses in excess of the deductible.
So, the family's out-of-pocket medical expenses for 2013 will be :
[tex]500+500=1000[/tex] dollars
As the limit was up to 4 deductibles in a calendar year, and in 2013, there were 2 claims, so that sums up to be $1000.
The family's out-of-pocket medical expenses for 2013 would be $1000, as they paid the $500 deductible for each of the two claims made that year, with their health insurance policy limiting to four deductibles per year.
Explanation:The subject of the question involves calculating the out-of-pocket medical expenses for a family under their health insurance policy, which includes understanding how deductibles work. In the scenario given, the family purchased a policy with a $500 deductible and a limit of four deductibles per calendar year. In 2013, they made two claims where each exceeded the deductible amount. Therefore, their out-of-pocket expenses for 2013 would be two times the deductible amount, since the policy has a limit of four deductibles per year but only two claims were filed and paid within that year.
Mathematically, this can be calculated as:
Claim 1 in September 2013: $500 (deductible)Claim 2 in September 2013: $500 (deductible)Total out-of-pocket expenses for 2013: $500 + $500 = $1000.
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A random sample of 10 chocolate energy bars of a certain brand has, on average, 230 calories per bar, with a standard deviation of 15 calories. Construct a 99% CI for the true mean calorie content of this brand of energy bar. Assume that the distribution of the calorie content is approximately normal.
Answer:345
Step-by-step explanation:
Final answer:
To construct a 99% confidence interval for the calorie content of the energy bars, calculate the standard error, find the appropriate t-value, then compute the margin of error, and add and subtract it from the sample mean. The resulting 99% CI for the true mean calorie content is approximately (214.59, 245.41) calories.
Explanation:
To construct a 99% confidence interval (CI) for the true mean calorie content of the chocolate energy bars, we will use the sample mean, the sample standard deviation, and the t-distribution since the sample size is small. Given are the sample mean (×) is 230 calories, the sample standard deviation (s) is 15 calories, and the sample size (n) is 10.
Steps to follow:
Identify the appropriate t-value for the 99% CI, which corresponds to a two-tailed test with 9 degrees of freedom (n-1). From the t-distribution table, this value is approximately 3.25.
Calculate the standard error (SE) of the mean by dividing the standard deviation by the square root of the sample size: SE = s / √n = 15 / √10 ≈ 4.74.
Multiply the t-value by the SE to get the margin of error (ME): ME = t * SE ≈ 3.25 * 4.74 ≈ 15.41.
Finally, subtract and add the ME from the sample mean to get the lower and upper bounds of the CI: (× - ME, × + ME) = (230 - 15.41, 230 + 15.41) = (214.59, 245.41).
Therefore, the 99% confidence interval for the true mean calorie content is approximately (214.59, 245.41) calories.
Explanation of a 95% CI: A 95% confidence interval means that if we were to take 100 different random samples from the population and construct a CI for each using the same method, approximately 95 of these intervals would contain the true population mean.
Clara has driven 70,000 miles in her car. On average, she drives 26 miles every day. Write a rule that represents her miles driven m as a function of time d.
Answer:
[tex]m(d)=26\ d[/tex]
where [tex]m[/tex] represents miles driven
and [tex]d[/tex] represents number of days driven.
Step-by-step explanation:
Given:
Total distance driven in miles = 70,000
Average distance driven each day = 26 miles
Taking average as unit rate of miles covered per day.
∴ we can say the car covers 26 miles per day.
Using unitary method to find miles driven in [tex]d[/tex] days.
In 1 day miles driven = 26
In [tex]d[/tex] days miles driven = [tex]26\times d =26\ d[/tex]
So, to find [tex]m[/tex] miles driven the expression can be written as:
[tex]m(d)=26\ d[/tex]
For each cost function (given in dollars), find (a) the cost,average cost, and marginal cost at a production level of 1000units; (b) the production level that will minimize the averagecost; and c) the minimum average cost.C(x)= 16,000x + 200x+ 4x3/2
Answer:
a) $342,491
$342.491
$389.74
b) $400
c) $320
Step-by-step explanation:
the cost function = C(x)
C(x) = 16000 + 200x + 4x^3/2
a) when we have a unit of 1000 unit, x= 1000
C(1000) = 16000 + 200(1000) + 4(1000)^3/2
= 16000 + 200000 + 126491
= 342,491
Cost = $342,491
Average cost= C(1000) / 1000
= 342,491/1000
= 342.491
The average cost = $342.491
Marginal cost = derivative of the cost
C'(x) = 200 + 4(3/2) x^1/2
= 200 + 6x^1/2
C'(1000) = 200 + 6(1000)^1/2
= 389.74
Marginal cost = $389.74
Marginal cost = Marginal revenue
C'(x) = C(x) / x
200 + 6x^1/2 = (16000 + 200x + 4x^3/2) / x
200 + 6x^1*2 = 16000/x + 200 +4x^1/2
Collect like terms
6x^1*2 - 4x^1/2 = 16000/x + 200 -200
2x^1/2 = 16000/x
2x^3/2 = 16000
x^3/2 = 16000/2
x^3/2 = 8000
x = 8000^2/3
x = 400
Therefore, the production level that will minimize the average cost is the critical value = $400
C'(x) = C(x) / x
C'(400) = 16000/400 + 200 + 4(400)^1/2
= 40 + 200 + 80
= 320
The minimum average cost = $320
Give the coordinates for the image of R (0, -5) E (4, -3) C (6, -5) T (2, -7) when it is reflected across the line y = x.
Answer:
Below.
Step-by-step explanation:
The x and y coordinates flip with this reflection e, g (2, 1) goes to (1, 2).
So R (0, -5) ---> R' (-5, 0).
E (4, -3) ---> E' (-3, 4).
C (6, -5) ---> C' (-5, 6).
T (2, -7) ---> T' (-7, 2).
Find the area of a triangle with the given vertices.
Part I: Graph the following points on the coordinate grid below.
(1, -3), (3, -1), (5, -3)
Part II: Find the area of the triangle. Show your work.
Answer:
the area of the triangle is 4 square units.
Step-by-step explanation:
Plotting the points, we can see that the triangle is isosceles lying in the 4th quadrant of graph.
we can break the triangle in 2 similar right angled triangles,
each with base 2 and height 2 units.
area of triangle is given by the formula,
A= [tex](\frac{1}{2})(base)(height)[/tex]
thus, A= [tex](\frac{1}{2})(2)(2)[/tex]
A=2 square units.
there are 2 such triangles,
thus total area is 4 square units.
A local hamburger shop sold a combined total of 498 hamburger and cheeseburger on Sunday. 52 fewer cheeseburgers sold than hamburgers how many hamburgers were sold on Sunday
Final answer:
Upon setting up an equation with the given information, we find that the shop sold 275 hamburgers on Sunday.
Explanation:
To find out how many hamburgers were sold on Sunday when the hamburger shop sold a combined total of 498 hamburgers and cheeseburgers, we can set up an equation. Let's denote the number of hamburgers as H and cheeseburgers as C. We are given that there were 52 fewer cheeseburgers sold than hamburgers, so we can express this as C = H - 52.
Since the total number of burgers sold was 498, we can also set up the following equation: H + C = 498. Substituting for C, we get H + (H - 52) = 498. Solving this equation, we get 2H - 52 = 498. Adding 52 to both sides gives us 2H = 550, and dividing by 2 gives us H = 275.
Therefore, the shop sold 275 hamburgers on Sunday.
Dwight and Walt are building model cars. Dwight builds 7 fewer models than 4 times the number Walt builds.Dwight builds at most 9 models.Which inequality could be used to find the number of models Walt builds.
Answer: w lesser than or equal to 4
Step-by-step explanation:
Dwight and Walt are building model cars.
Let d = the number of models built by Dwight.
Let w = the number of models built by Walt.
Dwight builds 7 fewer models than 4 times the number Walt builds. This can be expressed as
d = 4w - 7 - - - - - - - - - 1
Dwight builds at most 9 models. This is expressed as
d lesser than or equal to 9
From equation 1
d = 4w - 7
4w = d + 7
w = (d+7)/4
Assuming Dwight built 9 models
w = (9+7)/4 = 4
Therefore,
Walt builds at most 4 models. It is expressed as
w lesser than or equal to 4
It is shown in the attached photo
Answer:
lesser than or equal to 4 i think i tried my hardest sorry if its wrong
Step-by-step explanation:
Solve for (c).
12c−4=14c−10
c= ?
Answer:
C=3
Step-by-step explanation:
12c-4=14c-10 Given
6=2c Add 10 and subtract 12 from both sides
c=3 Divide by 2 to isolate the c
The value of c is 3 in the equation 12c−4=14c−10.
The given equation is 12c−4=14c−10
Twelve times of c minus four equal to forteen times c minus ten.
We have to find the value of c.
c is the variable in the equation.
Take the variable terms on one side and constants on other side.
12c-14c=4-10
-2c=-6
Divide both sides by 2:
c=3
Hence, the value of c is 3 in the equation 12c−4=14c−10.
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A submarine let Hawaii two hours before an aircraft carrier. The vessels traveled in opposite directions. The aircraft carrier traveled at 25 mph for nine hours. After this time the vessels were 280 miles apart. Find the submarines speed.
Answer: the speed of the submarine is 5miles per hour
Step-by-step explanation:
The submarine left Hawaii two hours before the aircraft carrier.
Let x = the speed of the submarine
The aircraft carrier traveled at 25 mph for nine hours.
After this time the vessels were 280 miles apart. This means that when they became 280 miles apart, the aircraft carrier has travelled for 9 hours. If the submarine was ahead of the aircraft carrier with 2 hours, that means that the submarine travelled 9 + 2 = 11 hours
Distance travelled = speed × time
Distance travelled by submarine will be 11 × x = 11x miles per hour
Distance travelled by aircraft carrier will be 25 × 9 = 225 miles per hour
If they are 280 miles apart, this would be their total distance. Therefore,
225 + 11x = 280
11x = 280 - 225 = 55
x = 55/11 = 5miles per hour
To find the submarines speed, we can set up an equation using the given information and solve for the unknown variable. The speed of the submarine is found to be 5 mph.
Explanation:To solve this problem, we need to set up an equation using the information given. Let's denote the speed of the submarine as 's'. The submarine traveled for two hours longer than the aircraft carrier, so the total time traveled by the submarine is '9 + 2 = 11' hours. The total distance between the vessels is given as 280 miles.
To find the speed of the submarine, we can use the formula: Distance = Speed * Time. Plugging in the given values, we can write the equation as: 280 = (25 mph * 9 hours) + (s mph * 11 hours).
Simplifying the equation gives us: 280 = 225 + 11s. Subtracting 225 from both sides gives us: 55 = 11s. Dividing both sides by 11 gives us: s = 5.
Therefore, the speed of the submarine is 5 mph.
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After a storm damages the community center, Shanika and her friends hold fundraising events to help pay for repairs. After the first event, they raise $240, which is 10% of the total amount that they want to raise. What is the total amount of money that Shanika and her friends want to raise?
Answer:Shanny and her friends wanted to raise $2400
Step-by-step explanation:
Fundraising events were held by Shanika and her friends to help pay for repairs.
Let x = the total amount of money that Shanika and her friends want to raise during the fund raising events. After the first event, they raise $240,which is 10% of the total amount that they want to raise. This means that after the first event, they raised 10/100 ×x = 0.1x
This 0.1x that they raised is equal to $240. Therefore,
0.1x = 240
x = 240/0.1 = 2400
Shanny and her friends wanted to raise $2400
Answer:
$2,400 is the correct answer
Step-by-step explanation:
can anyone help me? I've been stuck with this problem for hours
Answer:
336.02 square centimeters
Step-by-step explanation:
The surface area is the area of all the surfaces of the prism shown.
The prism has 7 surfaces.
Top and Bottom are pentagons with side lengths of 5.
The other 5 side surfaces are rectangles with length 10 and width 5.
Note the formulas of area of pentagon and area of rectangle below:
Area of Rectangle = Length * Width
Area of Pentagon = [tex]\frac{1}{4}\sqrt{25+10\sqrt{5} }* a^2[/tex] , where a is the side length
Lets find area of each of the surfaces:
Top Surface (Pentagon with side length 5) = [tex]\frac{1}{4}\sqrt{25+10\sqrt{5} }* a^2=\frac{1}{4}\sqrt{25+10\sqrt{5} }* (5)^2=43.01[/tex]
Bottom Surface = same as Top Surface = 43.01
Side Surface (rectangle with length 10 and width 5) = 10 * 5 = 50
There are 5 side surfaces that are each 50 sq. cm. so area would be:
Area of 5 Side Surface = 5 * 50 = 250
Total Surface Area = 250 + 43.01 + 43.01 = 336.02 square centimeters
The diameter of a cylindrical construction pipe is 5 ft. If the pipe is 21 ft long, what is its volume?
Use the value 3.14 for 7, and round your answer to the nearest whole number.
Be sure to include the correct unit in your answer
The volume of the cylindrical pipe is approximately 412 cubic feet.
Given:
- Diameter of the pipe = 5 ft
- Length of the pipe = 21 ft
First, we need to find the radius r of the cylinder. The diameter is 5 ft, so the radius is half of that, which is [tex]\( \frac{5}{2} = 2.5 \)[/tex] ft.
[tex]\[ \text{Volume} = \pi r^2 h \][/tex]
Substituting the given values:
[tex]\[ \text{Volume} = 3.14 \times (2.5)^2 \times 21 \]\[ \text{Volume} = 3.14 \times 6.25 \times 21 \]\[ \text{Volume} = 3.14 \times 131.25 \]\[ \text{Volume} \approx 412.425 \, \text{cubic feet} \][/tex]
Rounding to the nearest whole number, the volume of the cylindrical pipe is approximately 412 cubic feet.
What is the minimum number of students, each of whom comes from one of the 50 states, who must be enrolled in a university to guarantee that there are at least 100 who come from the same state?.
Answer:
Minimum number of student is 4951
Step-by-step explanation:
4950 wont work because there are 99 student in each state
99 *50 =4950
there are 100 students comes from same state. So from pigeon hole principle there are at least [ 4951/50] = 100 come from state
orest fire covers 2008 acres at time t equals 0. The fire is growing at a rate of 8 StartRoot t EndRoot acres per hour, where t is in hours. How many acres are covered 24 hours later? Round your answer to the nearest integer.
Answer: There are 2635 acres covered 24 hours later.
Step-by-step explanation:
Since we have given that
At time t = 0, number of acres forest fire covers = 2008 acres
We first consider the equation:
[tex]A=\int\limits^t_0 {8\sqrt{t}} \, dt\\\\A=8\dfrac{t^{\frac{3}{2}}}{\frac{3}{2}}+C\\\\A=\dfrac{16}{3}t^{\frac{3}{2}}+C[/tex]
At t=0, A= 2008
So, it becomes,
[tex]2008=C[/tex]
So, now it becomes,
[tex]A=\dfrac{16}{3}t^{\frac{3}{2}}+2008\\\\At\ t=24,\\\\A=\dfrac{16}{3}(24)^{\frac{3}{2}}+2008\\\\A=2635.06[/tex]
Hence, there are 2635 acres covered 24 hours later.
There were 90 people at a party. There were four more men than women and there were 10 more children than adults now many men women and children were at the party?
Answer:
288
Step-by-step explanation:
Answer: 50 children
18 women
22 men
Step-by-step explanation:
There were 90 people at a party. The persons consists of men, women and children. The men and women are adults.
Let m = number of men at the party
Let w = number of women at the party
Let c = number of children at the party
There were four more men than women. It means
w = m - 4
There were 10 more children than adults. It means
m + w + 10 = c
c = m + w + 10
Substituting w = m- 4 into the above equation, it becomes
c = m + m- 4 + 10 = 2m+ 6
Note: adults = sum of men and women
There were 90 people at a party. It means
m + w + c = 90
Substituting c = 2m+6 and w = m-4, it becomes
m + m-4 + 2m+6 = 90
4m = 90 - 18 = 88
m = 88/4= 22
w = m- 4 = 22-4
w = 18
c = 2m + 6 = 44 + 6 = 50
c = 50
The half-life of a certain substance is 26 years. How long will it take for a sample of this substance to decay to 92% of its original amount? Use the exponential decay model, A = A_0 e kt, to solve. years (Round to one decimal place as needed.)
Answer:
t= 3.1 years
Step-by-step explanation:
A = A_0 e kt
Half life(1/2) = 26 yrs
1/2 = 1_0 e^k.26
ln(1/2) = ln(e^26k)
26k. ln(e) = ln(1/2)
k = 1/26* ln(1/2)
k = -0.0267
A = A_0 e^kt
0.92 = 1.e^(-0.0267)t
ln(0.92) = ln(e^(-0.0267)t
-0.0267t .ln(e) = ln(0.92)
t = ln(0.92) / -0.0267
t = 3.122
t = 3.1years (approximate to 1 d.p)
Final answer:
The half-life of a substance is used to calculate how long it will take for a certain amount of it to decay. In this case, it will take approximately 3.2 years for the sample to decay to 92% of its original amount using the given half-life of 26 years and the exponential decay model.
Explanation:
The half-life of a substance is the time it takes for half of it to decay. Given that the half-life of a certain substance is 26 years, we can use the exponential decay model A = A0ekt, where k is the decay constant. To solve for the remaining 92% of the substance, we would set A to 0.92A0. The decay constant k is related to the half-life (t1/2) by the equation k = -ln(2) / t1/2. So, let's solve for k and then use it to find the time (t) it takes for the sample to decay to 92% of its original amount.
First, find the decay constant k using the half-life:
k = -ln(2) / 26 yrs = -0.0267 per year (rounded to four decimal places)
Now, set up the equation:
0.92A0 = A0e(-0.0267)t
Divide both sides by A0 and take the natural logarithm:
ln(0.92) = -0.0267t
Solving for t gives:
t = ln(0.92) / -0.0267 ≈ 3.2 years (rounded to one decimal place)
It will take approximately 3.2 years for the sample to decay to 92% of its original amount.
A boat leaves New Orleans and travels up stream on the Mississippi River for 4 hours the return trip takes only 2.8 hours because the boat travels 3 miles per hour faster downstream due to the current. How far does the boat travel up stream?
Answer: 3.73
Step-by-step explanation:
In parallelogram ABCD
What is BD
Answer: BD = 108
Step-by-step explanation:
In a parallelogram, the opposite sides are congruent and the diagonals bisect each other. It means that they bisect at a midpoint that divides them equally.
Therefore,
AB = DC
AD = BC
BD = AC
Also BE = ED. This means that
7x - 2 = x^2 - 10
x^2 - 10 +2 - 7x = 0
x^2 - 7x -8 = 0
Solving the quadratic equation with factorization method,
x^2 + x - 8x -8 = 0
x(x + 1) -8(x + 1) = 0
x - 8 = 0 or x + 1 = 0
x = 8 or x = -1
Since x cannot be negative,
x = 8
BE = 7×8 - 2 = 54
ED = 8^2 - 10 = 54
BD = BE + ED = 54 +54 = 108
A flower vase has 5 white lilies, 4 pink roses, and 6 yellow carnations. One flower is chosen at random and given to a woman for her to keep. Another flower is then chosen at random and given to a different woman for her to keep. Both women received a pink rose. Are these events independent or dependent
Answer: These events are dependent.
Step-by-step explanation: The probability of the second woman getting a pink rose is affected by the first woman getting a pink rose as the pink rose obtained by the first woman was not replaced. Hence there are less pink roses in the flower vase and hence lower probability that the second woman gets a pink rose. These events are thus dependent.
Answer:
Dependent because when the first flower is taken, it affects the ratio of the types of flowers in the vase.
Step-by-step explanation:
The chance of A is 1/3; the chance of B is 1/10. True or false, and explain: (
a) If A and B are independent, they must also be mutually exclusive.
(b) If A and B are mutually exclusive, they cannot be independent.
Answer:
a) False
b) True
Step-by-step explanation:
Given : The chance of A is [tex]\frac{1}{3}[/tex]; the chance of B is [tex]\frac{1}{10}[/tex]
To find : True or false, and explain ?
Solution :
a) If A and B are independent, they must also be mutually exclusive.
Two events are mutually exclusive, if the events cannot occur at the same time.
When events A and B are independent, then the chance of event B is not affected by event A occurring.
However, when events A and b are mutually exclusive, then the chance of event B needs to change to 0 when event A has occurred as B cannot occur when A occurs.
Which means the given statement is false.
(b) If A and B are mutually exclusive, they cannot be independent.
Two events are independent, if the probability that one event occurs in no way affect the probability of the other event occurring.
When events A and B are mutually exclusive, then the chance of event B needs to change to 0 when event A has occurred as B cannot occur when A occurs.
Which means, the given statement is true.
It is false that independent events must be mutually exclusive, as independence indicates no effect on each other's occurrence, not impossibility of simultaneous occurrence. Conversely, if two events are mutually exclusive, they cannot be independent because the occurrence of one negates the possibility of the other, affecting the probabilities.
Explanation:The question pertains to the concepts of independent and mutually exclusive events in probability theory. Here we are presented with two parts:
(a) If A and B are independent, they must also be mutually exclusive. This statement is false. If events A and B are independent, this means the occurrence of A does not affect the occurrence of B and vice versa. Meanwhile, mutually exclusive events cannot happen at the same time. Therefore, independence does not imply mutual exclusivity.(b) If A and B are mutually exclusive, they cannot be independent. This statement is true. If events A and B are mutually exclusive, it means that if one occurs, the other cannot, hence they are not independent. Independence implies that the occurrence of one event does not affect the probability of the other, which contradicts mutual exclusivity.To further clarify, independence between two events A and B is defined mathematically as P(A and B) = P(A)P(B). Meanwhile, two events are mutually exclusive if P(A and B) = 0.
A or B or C or D which expression??
Answer:
option A
Step-by-step explanation:
Notice that you need to emulate the series: 1 + 5 + 25 + 125 + 625 (a five total term series)
with the indicated sums.
The first term in the your series (addition) has to be "1". This fact already gets rid of two of the suggested sums (B, and D) because their first term is [tex]5^1=5[/tex].
So, now analyzing the options A and C, we notice that A has a sum from i=0 to 4 (which gives a total of five terms ao, a1, a2, a3, and a4, while option C has a total of six terms (from i = 0 to 5): a0, a1, a2, a3, a4, a5.
S, the obvious candidate is option A. So now evaluate the five terms corroborating that:
[tex]5^0 + 5^1+5^2+5^3+5^4=\\=1+5+25+125+625[/tex]
Therefore, option A is the answer