A water tank is in the shape of a right circular cone as shown above. The diameter of the cone is 10 feet, and the height is 15 feet. The shape of the water in the tank is conical with radius r feet and height h feet. At noon, water is leaking from the bottom of the tank at a rate of 12 cubic feet per hour, and the volume of water in the tank is 27π cubic feet. At noon, what is the rate at which the height of the water in the tank is changing? (The volume V of a right circular cone with radius r and height h is V=13πr2h.)

Answers

Answer 1

The rate at which the height of water tank changing is [tex]0.4244 \; \text {ft./hr}[/tex]

Step-by-step explanation:

Given information:

[tex]\text{\bold{The diameter of cone}}=10\;\rm{ ft.}[/tex]

[tex]\rm{The\; height\; of\; the\; cone}= 15\;\rm{ ft.}[/tex]

[tex]\text{The rate at which the water is leaking }= 12 \; \rm ft^3/hr.[/tex]

[tex]\text{Volume of water tank}=27 . \pi \; \rm ft^3[/tex]

Now,

If ,we take a right circular cone with radius r and height h , its volume

[tex]\text{volume (V)} = (1/3) \times \pi \times \rm{r^3 \times h[/tex]

Now, the radius of cone when the volume of water tank is given

[tex]\frac{1}{3}\times \pi \rm\times{r^2 \times h = 27 \times \pi[/tex]

The ratio of height to radius of the cone :

[tex]\text {h/r} =3[/tex]

[tex]\text r = \text h/3[/tex]

Now calculate :

[tex]dV/dt= dV/dh \times dh/dt[/tex]

Which gives:

[tex]12 = \frac{d(\text h^3/27 \pi)}{dh} \times \frac{d\text h}{dt} \\\\\frac{d\text h}{dt} = \frac{12}{\pi \text h^2/9}[/tex]

As ,

[tex]27=\frac{1}{27 \times h}[/tex]

[tex]\text h^3=27^2\\\text h= 9\\[/tex]

Now,

[tex]dh/dt=12/(\pi.\text h^2/9)\\dh/dt=4/3\pi\\dh/dt=0.4244\\[/tex]

Hence, The rate at which the height of water tank changing is [tex]0.4244 \; \text {ft./hr}[/tex]

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Related Questions

Find the measures of the exterior angles of the polygon.

Answers

Step-by-step explanation:

180 - 34 = 146

one angle is a right angle, angle across it is 146, 360 - 90 - 146 = 124

divide that by two to get the measures of the remaining two angles (62 degrees)

180 - 62 = x

118 = x

:

/ = .

-- :

, . , - , ,

Im so confused pls help

Answers

Step-by-step explanation:

[tex] f(x) \: = \: \frac{4}{5} - \frac{5}{x + 1} [/tex]

[tex]f(x) \: \: = \: \: \frac{(4)(x + 1) - 5(5)}{5x + 5} [/tex]

[tex]f(x) \: \: = \: \: \frac{4x + 4 - 25}{5x + 5} [/tex]

[tex]f(x) \: \: = \: \: \frac{4x - 21}{5x + 5} [/tex]

f(x) = g(x)

[tex] \frac{4x - 21}{5x + 5} = \frac{2}{5x + 5} [/tex]

4x - 21 = 2

x = (21 + 2)/4

x = 5.75

the diagram shows a circle inside a square ABCDB AB = 20 cm. the radius of the circle is 9 cm. work out the area of the shaded region

Answers

Step-by-step explanation:

Area of the square = a²

= 20²

= 400cm²

Area of the circle = πr²

= 22/7 × 9²

= 22/7 × 81

= 254.57cm²

Area of the shaded region = Area of the square - Area of the circle

= 400cm² - 254.57cm²

= 145.43cm²

Tyler went to the supermarket to buy food for a food pantry. He has $36, and can carry up to 20 pounds of food in his backpack.

Pasta costs $1 for a 1-pound package. Pasta sauce costs $3 for a 1.5 pound jar.

Let x = the number of packages of pasta and y = the number of jars of pasta sauce.

One package of pasta is the right amount to go with one jar of pasta sauce. What is the best numbers of packages of pasta and jars of pasta sauce to buy for the food pantry?

How many packages of pasta?

How many jars of pasta sauce?

Explain your reasoning.

Answers

Answer:

8 packages of pasta and 8 jar of sauce which is $32 dollars and 20lbs in total

Step-by-step explanation:

8 pack of pasta is 8$ and 8lbs

8 jars of sauce is 24$ and 12lbs

8 plus 12 is 20 which is the max of lbs Tyler can carry.

Challenger Elementary School has 800 students. Every Wednesday, 12% of the students stay after school for Chess Club. How many students attend on Wednesday’s?

Answers

Answer:

96 students attend chess club after school on Wednesday's.

Step-by-step explanation:

To solve this problem, simply multiply 800 x 0.12.

    800 x 0.12 = 96

I hope this was helpful to you! If it was, please consider rating, pressing thanks, and marking my answer Brainliest. It would help a lot. Have a great day!

Answer:

[tex]\huge\boxed{\text{96 students}}[/tex]

Step-by-step explanation:

In order to find how many students stay back for Chess Club, we want to find 12% of 800.

To do this, we can set up a percentage proportion.

[tex]\frac{12}{100} = \frac{x}{800}[/tex]

We can now solve for x by cross multiplying.

[tex]800 \cdot 12 = 9600\\9600 \div 100 = 96[/tex]

So, 96 students attend Chess Club on Wednesdays.

Hope this helped!

Stan, Greg, and Seth were cleaning the school gym in shifts. Stan
cleaned first, then Greg, then Seth. If they each cleaned the same
amount of time for a total of 11 hours, how many minutes did each one
clean?

Answers

an hour is 60 minutes so we would do 60x11 which equals 660 then divide by 3 which is 220 so 220 minutes is your answer

2+10= 12

True or False??

Just for points:)

Answers

Answer:

True thank you for the points

Answer:

true

Step-by-step explanation:

2+10=12

true. do i get the points?

A person’s weekly pay is directly proportional to the number of hours worked. Shawn’s pay is $123.00for 20 hours of work. Find the amount of pay for 31 hours of work. ONLY NUMERICAL VALUE

Answers

Answer:

$190.65 for 31 hours

Step-by-step explanation:

P= ha

123=20a

123/20=a

6.15=a

P=6.15h

P= 6.15(31)

p=190.65

Solve the right triangle

Answers

Answer:

JK = 50; IJ = 25√3

Step-by-step explanation:

HAVE: sin ∠J = sin 30° = [tex]\frac{IK}{JK}[/tex]

=> [tex]\frac{IK}{JK}=\frac{25}{JK}=\frac{1}{2}=> JK = 25.2 = 50[/tex]

cos ∠J = cos 30° = [tex]\frac{IJ}{JK}[/tex]

=> [tex]\frac{IJ}{JK}=\frac{IJ}{50}=\frac{\sqrt{3} }{2} => IJ = \frac{\sqrt{3} }{2}.50=25\sqrt{3}[/tex]

Your cell phone plan costs $34.99 per month plus $0.15 for each text message you send or receive. You have at most $40 to spend on your cell phone bill. what is the maximum number of text messages that you can send or receive next month?

Answers

Answer:

33 texts

Step-by-step explanation:

34.99+0.15(t)=40

lets say t is number of texts

subtract 34.99 from both sides

0.15(t)=$5.01

divide 5.01 by .15

33.4

but you cant send .4 of a text

so 33 texts

find the x and y intercept
y=5x-13

Answers

Answer:

X int = 13/5    Y int= -13

Step-by-step explanation:

Suzy ordered some games online for $19 each. She paid a total of $8 for shipping. The total cost of the purchase was $122. How many games did she buy? 7) Using the word problem above, write an equation to find the number of games Suzy bought. (Hint: Don't do any of the steps to solve it, just write the equation) Your answer​

Answers

Answer:

6

Step-by-step explanation:

122-8 shipping =114 /19=6

Find the third side in simplest radical form:
1&3

Answers

Answer:

[tex]\sqrt{10}[/tex]

Step-by-step explanation:

[tex]1^{2} + 3^{2} =x^{2} \\1+9=x^{2} \\10=x^{2} \\\sqrt{10} =x\\[/tex]

PLEASE HELP ME AND EXPLAIN.

Answers

Answer:

9*8=72 which would be more than enough to cover the wall

Find the zero of the function.
f(x) = 4x2 – 12x + 9

The zero of the function is x=

Answers

Answer:

x= 3/2

Step-by-step explanation:

I apologize if it is wrong!

A student wrote four statements during math class. Which statement contains an error?

Answers

Answer:

C

Step-by-step explanation:

The answer is C if you multiply (a)38×18=684

(b)46×16=736 (c)54×14=756 the question is 766 miles meaning that c is the wrong answer .i hope this was helpful.

9. a - 2a +5
pls help i’m kinda dumb

Answers

a-2a +5= -a + 5 would be the answer
Answer: -a+5
First you combine like terms
a-2a equals -a
After that you can’t combine anything else
So the equation simplified is -a+5

I need help on this one

Answers

Answer:

A

Step-by-step explanation:

(-2,11) (4,-7) —> y2-y1/x2-x1= m. Then to find b take (4,-7) and plug those numbers into y=Mx+b

Find 0.038 as a fraction in its lowest terms

Answers

Answer:

19 / 500

Pls mark as brainliest!!!

Answer:

19/5

Step-by-step explanation:

If you're using a scientific calculator you will be able to find this answer by just simply pressing 0.038 x 100 divided by 1

PLEASE help find measure of angle - TEst review today

Answers

Answer:

both measured would be 25 and 19

Step-by-step explanation:

Answer:

x=4

Step-by-step explanation:

if x equals 4 then 21x+4 equals 88 and 23x-4 equals 88 too.

Find x (Type the number answer)

Answers

He’s with his other girl TvT

Answer:

18.41

Step-by-step explanation:

[tex]x^{2} = 12^{2} + 8^{2} - 2(12)(8)(cos 133)\\\\x= \sqrt{338.94} = 18.41[/tex]

what is -2(15 ÷ 5) + 7(-2)

Answers

Answer:

-20

Step-by-step explanation:

Using the divisibility rules, explain why 678 is divisible by 2, 3, and 6.

Answers

Answer:

678

678 is even, so it's divisible by 2

6+7+8 = 21, which is divisible by 3, so 678 is divisible by 3

Since 678 is divisible by both 2 and 3, it has to be divisible by 6

Step-by-step explanation:

Which rational number equals 0.7 ? 1 10 Flo 11​

Answers

Answer:

0.7 is a rational number... but 7/10

Step-by-step explanation:

Hope I helped!

Answer: This absolutely makes no sense  

Step-by-step explanation:

HELP ME PLEASE!
Complete the work to find the equation of a line that passes through point (–5, –3) and is parallel to line y – 3x = –7. Step 1: Write the equation of the given line in slope intercept form: y = 3x – 7. Step 2: Determine the slope of the parallel line: m = 3. Step 3: Substitute the slope and given point into y = mx + b to find the y-intercept of the new line. –3 = 3 (–5 ) + b 3 = –15 + b b = 12 Step 4: Write the equation of the line parallel to 3x – y = 7 that passes through point (–5, –3).

Answers

Answer:

Y=3x+12 (its the first option)

Step-by-step explanation:

I did the assignment.

The equation of the line parallel to (3x - y = 7) passes through the point (-5, -3) is y = 3x + 2 and this can be determined by using the one-point slope form.

Given :

The line passes through the point (-5, -3) and is parallel to the line (y - 3x = -7).

The following steps can be used in order to determine the equation of a line that passes through the point (-5, -3) and is parallel to the line (y - 3x = -7):

Step 1 - The one-point form of a line is used in order to determine the equation of a line that passes through the point (-5, -3) and is parallel to the line (y - 3x = -7).

Step 2 - The one-point form is given below:

[tex](y-y_1) = m (x-x_1)[/tex]

Step 3 - According to the given data, the line is parallel to the line (y = 3x - 7). So, the slope of the line is 3.

Step 4 - Now, substitute the values of slope and the point in the above expression.

[tex](y+3) = 3 (x+5)[/tex]

y = 3x + 2

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Make & the subject of the
Formula
6x ta=5(x+t)

What is the answer please

Answers

Answer is x = 5t-a

6x+a = 5(x+t) Expand the bracket
6x+a = 5x+5t Subtract the least x’s (so 5x)
x+a = 5t Subtract a to make x the subject
x = 5t-a

Answer:

x  = 5t - a

Step-by-step explanation:

The problem deals with making a variable the subject of the formula in the expression.

 Given expression:

            6x + a = 5(x + t)

  We are to make x the subject of the formula in the expression

   

Solution:

           6x + a  = 5(x+t)

On the right hand side, expand the parentheses and distribute the term outside;

          6x + a  = 5x + 5t

   Bring the term desired to be the subject to one side by adding (-5x) to both sides;

     6x + (-5x) + a = 5x + (-5x) + 5t

           x + a  = 5t

  Now add (-a) to both sides

         x + a + (-a) = 5t + (-a)

          x  = 5t - a

HURRRRY!!!! Thank you

Answers

Answer:

C

Step-by-step explanation:

Uhm I tried my best im so sorry if it's wrong

it is B yep he was right have a nice day

HELP:( I know I’m slow and all asking all these question today but...

Answers

Second answer

Step-by-step explanation:

Select all that apply.
40 is what percent of 25. Hint: There are 2 correct answers.
А160%
B 62.5%
C 1600%
D625/1000
E 160/100
F1600/100

Answers

Answer:

B and D

Step-by-step explanation:

Hope this helps :3

if it did pls mark brainliest

Answer:

A & E

Step-by-step explanation:

First, multiply 40 by 100, giving you 4000. You then divide 4000 by 25, where you will get a quotient of 160. Now, this can be represented as 160% or 160/100 because a percentage is always out of 100.

How many modes, if any, does the group of data have?
Kate, Tim, John, John, Angela, Tim, Sue 1 mode 2 modes no mode 3 modes

Answers

The number of modes in the dataset is 1. The dataset is unimodal. The correct option is therefore;

1 mode

What is a unimodal data?

A unimodal data is a set of data that has only one mode. The mode is a measure of central tendency.

The data in the question can be presented as follows;

Kate, Tim, John, John, Angelam, Tim, Sue

The frequency of each of the data is therefore;

Kate; 1, Tim; 1, John; 2, Angela; 1, Tim; 1, Sue; 1

The mode of a dataset, is the value that has the highest frequency, or that appears the most in the dataset.

The data that has the highest frequency is John, the mode of the dataset is therefore, John, which has a frequency of 1

The dataset, therefore, has only 1 mode.

Learn more on the measures of central tendency, mode median, and mean, of a dataset here: https://brainly.com/question/24994390

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