Question

Suppose that the length l, width w, and perimeter P of a rectangle are differentiable functions of t. Suppose also that the length is always three times and twenty-two more than the width. Write an equation that relates dPdt to dwdt. Enter your answers using exact values.

Answer 1

Answer:

$\dfrac{dP}{dt}=8\dfrac{dw}{dt}$

Step-by-step explanation:

The perimeter, P, is given by

$P(t) = 2(l(t)+w(t))$

Since $l(t) = 3w(t) + 22$,

$P(t) = 2(3w(t) + 22+w(t)) = 2(4w(t)+22) = 8w(t)+44$

Differentiating P with respect to w,

$\dfrac{dP}{dw} = 8$

But

$\dfrac{dP}{dw} = \dfrac{dP}{dt}\times \dfrac{dt}{dw} = \dfrac{dP}{dt}/\dfrac{dw}{dt}$

$\dfrac{dP}{dt} = \dfrac{dP}{dw} \times \dfrac{dw}{dt} = 8\dfrac{dw}{dt}$