A manufacturing process produces two types of metal rods whose diameters (in…

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

A manufacturing process produces two types of metal rods whose diameters (in millimeters) measured at position

x

centimeters from one end are modeled for

0\le x\le10

r_1(x)=10+0.4x\sin\!\left(\frac{\pi x}{5}\right)

r_2(x)=10+kx^2\cos\!\left(\frac{\pi x}{10}\right),

where

k

is a real constant. Quality control requires that the average diameter of type 2 rods not exceed the average diameter of type 1 rods over the

10

-cm length. (a) Derive an expression for the average diameter of each rod type on the interval

[0,10]

in terms of

k

(if necessary). You may evaluate any definite integrals that arise. (b) Determine all real values of

k

for which the quality-control requirement is satisfied. Show complete analytical justification. (c) For the smallest positive

k

found in part (b), interpret how the functional form of

r_2

causes its average diameter to match the requirement while still differing from

r_1

at specific positions along the rod.