Name Math 124 - Test 3 Take home portion You must show all of your work to recei

Question

Name Math 124 - Test 3 Take home portion You must show all of your work to receive full credit and your work must clearly demonstrate the correctness of your answer. Your work must also be neat and organized. This portion of the exam is due on Monday, April 16. 1. (12 points) The combined surface area of R2-D2's hemispherical head and cylindrical body is 3648 em. Find the height of R2-D2 (i.e. R+D+4cm+R) given that his dimensions maximize the volume of his "body" (identified in the diagram below) where he keeps all of his gizmos and gadgets. Hemispherical Head 0 0808p Cylindrical Body

Explanation / Answer

SA = 2pi*r^2 + 2pi*r*h + pi*r^2

SA = 3pi*r^2 + 2pi*r*h

3pi*r^2 + 2pi*r*h = 3648pi

3r^2 + 2rh = 3648

So, h = (3648 - 3r^2) / (2r)

Now, we need to max volume....

So,
V = pi*r^2*h

V = pi*r^2*(3648 - 3r^2) / (2r)

V = (3648pi*r^2/(2r)) - 3pi*r^4/(2r)

V = 1824pi*r - 1.5pi*r^3

Now, for max V, dV/dr = 0 :
dV/dr = 1824pi - 4.5pi*r^2 = 0

4.5*pi*r^2 = 1824pi

r^2 = 405.3333333333333333

So, r = 20.133

And with tis,
h = 3648 - 3r^2) / (2r)

h = (3648 - 3*20.133^2) / (2*20.133)

h = 60.398

So, the height is :
20.133 + 60.398 + 20.133 + 4

104.664 cm ------> ANS

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