Relative rates question

Question

1. The length of a rectangle in increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width 10 cm, how fast is the area of the rectangle increasing?

Answer 1

L' = 8

W' = 3

L=20

W = 10

A= LW

product rule: 1st term times derivative of 2nd + 2nd term times derivative of 1st term

A' = LW' + WL'= 20(3) +(10)8 = 60+80 = 140 cm^2 per second

in one second, L increases from 20 to 28

in one second W increases from 10 to 13

in one second A increases from 200 to 364

364-200 = 164 = average rate of change over an interval of one second

140 = instantaneous rate of change of area

in the previous second L increased from 12 to 20

W from 7 to 10

and A from 84 to 200 = 116 rate of increase

(116+164)/2 = 280/2 = 140 cm^2 per second rate of change for Area

= average rate of change over a 2 second interval = instantaneous rate of change at the midpoint of that 2 second interval.

This is a check on the calculation for instantaneous rate of change of A