80 pi square inches of material of negligible thickness are required

Question

80 \pi  square inches of material of negligible thickness are required to construct a 1/16 scale model of a cylindrical barrel. If the diameter of the base of the barrel is 160 inches, then what is the volume of the barrel, to the nearest cubic foot?

A.  2\pi

B.  11\pi

C.  178\pi

D.  711\pi

E.  1280\pi

ruchik · Accepted Answer

Hungluu92vn   azoz7   Aim015   ayush19   stne  
Here is how i solved it:
Original barrel diameter = 160 inches, hence radius = 80 inches
Now we are told scaled model is 1/16 times original. Hence the radius of scale model is 80/16 = 5 inches.
Now surface area = 2(pi)r*h + 2*pi*r^2 = 80*pi
substitute r = 5 in above formula we get h = 3 inches.
Now this is the height of the model.
So height of original barrel is 3*16 = 48 inches.

Now we know height and radius of original barrel. so we can convert into feet, since final answer is required in feet.
Height = 48 inches = 4 feet
Radius = 80 inches = 20/3 feet.

Now volume of barrel = pi*r^2*h = pi*(20/3)^2*4 = (1600/9)*pi = ~178*pi
Hence answer is D

Please give kudos if you like the explanation.

Archit3110 · Answer

2*pi*r*(h+r)= 80*pi*16*16
solve for h = 48 inches = 4 feet
and radius = 80 inches ; 20/3 feet
so vol ; pi * r^2*h = pi*20*20/9 * 4 = 178 pi
IMO C

Fdambro294 · Answer

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

bumpbot · Answer

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80 pi square inches of material of negligible thickness are required

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80 pi square inches of material of negligible thickness are required

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