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As shown below, 7 circular pegs of the same radius are bundled together so that they are tangent to one another, and a string is wrapped around the outside. If the pegs' radii are all increased by 2 inches, how much longer would the string around the pegs be, in inches?

Your answer should be a decimal rounded to the nearest hundredth.

Mar 5, 2022

#2
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The string can be divided into two different parts, the straight part and the curving part (when tangent to a circle).

There are 6 straight parts, and each has a length of 2r. The total length is 2r * 6 = 12r.

The curving part adds up to the length of one of the circles, so its length should be $$\pi r^2$$.

Together the length is $$12r+\pi r^2$$.

If r is incremented by 2, we just have to replace r with (r+2). The length after incrementing is $$12(r+2)+\pi(r+2)^2$$.

Simplify, the length is $$\pi r^2+12r+4 \pi r +28$$.

The difference between the two lengths are $$(\pi r^2+12r+4 \pi r +28) - (\pi r^2+12r)=4 \pi r+28$$. And the string would be that much longer.

Mar 5, 2022