I'm not sure how to do this

Question

An industrial propane tank is formed by adjoining two hemispheres to the ends of a right circular cylinder. The length of the cylindrical portion of the tank is three times the radius of the hemispherical components . (Let L = 3r.)

(a) Write a function that represents the total volume V of the tank in terms of r.
V =


(b) Find the domain of the function.
r = 0
r ≤ 0
r ≥ 0
r ≥ 3


(c) Use a graphing utility to graph the function.


(d) The total volume of the tank is to be 120 cubic feet. Use the graph from part (c) to estimate the radius and length of the cylindrical portion of the tank. (Round your answers to two decimal places.) (In feet)

 

radius

length

Answer 1

If we take a sphere and cut it in half, we get a hemisphere.  If the volume of a sphere is (4/3)πr³, then the volume of a hemisphere is half the volume of a sphere.

 

Since we have two hemispheres, one at the one end of the cylinder and the other at the second end, it is like having a sphere in the cylinder.

 

 

a)

 

Volume of cylinder is   πr²L.

 

 

So the total volume of the tank is

 

V = πr²L + (4/3)πr³

 

Substitute  L=3r   into this formula.

 

 

V = πr²(3r) + (4/3)πr³

V = 3πr³ + (4/3)πr³

 

V = (13 / 3)πr³

 

 

b)

 

The domain is the values of r in which the function V exists.   If the radius is negative or zero, there is no shape.  So the domain in your answer choice that would make sense is

 

r ≥ 3

 

 

c)

 

In the graphing calculator, press Y= button.  Type the function of volume that we obtained in part a) into the calculator.  In the calculator, you would type (13/3)πX³.   Press GRAPH.  You will see the graph being constructed.

 

 

d)

 

Once your graph is constructed, press the TRACE button.  Press the left and right arrow keys on the calculator to move the cursor up the graph.  On the bottom of the screen, you will see x and y values changing as you move the cursor.  Move it to the point near y=120.   Read the x value near that point.  That will be your approximate radius.

 

Multiply the radius by 3 to get your length.