A cylindrical container for storing radioactive waste is to be constructed from lead and have a thickness of 6 in. (see the figure). If the volume of the outside cylinder is to be , find the radius and the height of the inside cylinder that will result in a container of maximum storage capacity. Hint: Show that the storage capacity (inside volume) is given by
A) r = ft.; h = 2 ft.
B) r = ft.; h = 3 ft.
C) r = ft.; h = 2 ft.
D) r = ft.; h = 3 ft.

Question

A cylindrical container for storing radioactive waste is to be constructed from lead and have a thickness of 6 in. (see the figure). If the volume of the outside cylinder is to be  , find the radius and the height of the inside cylinder that will result in a container of maximum storage capacity.  Hint: Show that the storage capacity (inside volume) is given by 
A) r = ft.; h = 2 ft. 
B) r = ft.; h = 3 ft. 
C) r = ft.; h = 2 ft. 
D) r = ft.; h = 3 ft.

Quizplus · Accepted Answer

Let's call the inside radius of the cylinder r and the inside height h. The outside dimensions of the cylinder will be (r+6) for the radius and (h+12) for the height (since there is 6 inches of lead on each end).

The volume of the container is given by:

V = π(r+6)^2(h+12) - πr^2h

Simplifying this expression, we get:

V = π(12rh + 72r^2 + 72rh + 432) - πr^2h

V = 432π + 84πrh + 72πr^2

To find the maximum value of V, we need to take the derivative and set it equal to 0:

dV/dr = 84πh + 144πr = 0

Solving for r, we get:

r = -7h/12

This means that V is at a maximum when r is negative. However, we know that r must be positive, so we take the positive value of r:

r = 7h/12

Now we need to find h. We know that the outside volume of the cylinder is given by:

V_outside = π(R^2)(H+12)

where R = r+6 and H = h+12.

We are given that V_outside = 11eaa8b1_db46_a86c_b1b6_05d076c3a97d_TB6026_11 , so we can plug in the values for R and H and solve for h:

11eaa8b1_db46_a86c_b1b6_05d076c3a97d_TB6026_11 = π(r+6)^2(h+12+12)

11eaa8b1_db46_a86c_b1b6_05d076c3a97d_TB6026_11 = π(7h/12+6)^2(h+24)

11eaa8b1_db46_a86c_b1b6_05d076c3a97d_TB6026_11 = 9144.02h + 12359.5

h = (11eaa8b1_db46_a86c_b1b6_05d076c3a97d_TB6026_11 - 12359.5)/9144.02

h = 2.998 ft.

Round this down to 3 ft. for h to get the final answer:

r = 7(3)/12 = 11eaa8b1_db47_1da0_b1b6_359605f04391_TB6026_11 ft.
h = 3 ft.

Therefore, the best choice is B.

A Cylindrical Container for Storing Radioactive Waste Is to Be