80 pi square inches of material of negligible thickness are required

Big trap on this question is not to get caught in the "Units Trap". The dimensions are all given in inches (cubic inches, etc.) but the answer asks for the Volume to the nearest cubic FOOT.

1 foot = 12 inches

the 80 (pi) is used to construct the Outer Surface Area of the 1/16th Model of the Barrel.


Step 1: Calculate the Surface Area of the 1/16th Model Barrel in order to find the Height = h

160 in = Actual Diameter

for the 1/16th model, every dimension will be scaled down by 1/16th
160 * 1/16 = 10 in = d

this means the radius of the Scale Model = r = 5

Surface Area of a Cylinder = (2 Areas of the Circle's Surface - the Top and Bottom of the Cylinder) + (Circumference of Circular Base * Height)

S.A. of Model = 80 (pi) = 2 * (pi) * (r)^2 + 2 * r * (pi) * h

Canceling the (pi) that appears in every term and Substituting r = 5 in for the Scale Model you get:

80 = 2 * (5)^2 + 2 * 5 * h
80 = 50 + 10h
30 = 10h
h = 3 inches ----- this is the Height of the Scale Model. So we need to Scale it UP for the Actual Sized Barrel 16 X Times.


Actual Barrel Height = h = 3 * 16 = 48 inches


Step 2: Biggest Trap - Convert the inches to feet and the Calculate the Actual Barrel's Volume

Actual Barrel Height = h = 48 inches

Actual Barrel Radius = r = 80 inches

Volume = (pi) * (r)^2 * h

Using Conversion Factors:
48 in * (1 foot / 12 in) = 4 feet = h in feet

80 in * (1 foot / 12 in) = 80/12 feet = 20 / 3 feet = r in feet

Plug h = 4 feet and r = 20/3 feet into the Volume Formula:

V = (pi) * (20/3)^2 * 4
V = (pi) * 400/9 * 4

since the Q asks for the nearest cubic foot, you can estimate 400/9 = approx. 44

V = (pi) * 44 * 4

V = 176 * pi

C is the Closest Answer

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