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.
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.