A roll of Scotch Tape has a diameter of 2.9 inches when new and 1.4 inches when used up. What is..

Question

A roll of Scotch Tape has a diameter of 2.9 inches when new and 1.4 inches when used up. What is the diameter when it is half used up? I know the answer, it's aprox 2.28 inches. But, I don't know why to use Linear Regression to get the answer, and also why it's not just the average of the two given diameters. Can you please explain why I would use Linear Regression instead of just averaging them please? Thank You!

Answer 1

the tape is wound around a small plastic cylinder

first find the area of the roll of tape (including the plastic cylinder)

the diameter is 2.9 inches so the radius is 1.45 inches

A=(pi)r^2

A=(3.14159)(1.45)^2

A=(3.14159)(2.1025)

A=6.60519 sq inches

now minus the area of the plastic cylinder

the diameter of the plastic cylinder is 1.4 inches so the radius is 0.7 inches

A=(3.14259)(0.7)^2

A=(3.14159)(0.49)

A=1.539379 sq inches

A=6.60519-1.53938

A=5.06581 sq inches of tape only

when the tape is half used up, you have 5.06581/2=2.532905 sq inches of tape

now add the area of the plastic cylinder to the area of the tape left over (which is half)

A=2.532905+1.539379

A=4.072284 sq inches

A=(pi)(r^2)

4.072284=(3.14159)(r^2)

r^2=4.072284/3.14159

r^2=1.296249

r=√1.296249

r=1.138529

diameter=2*radius

diameter=2*1.138529

diameter=2.277 which is approximately 2.28 inches