question suppose that the length l, width w, and perimeter p of a rectangle are differentiable functions of…

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 eighteen more than the width. write an equation that relates dp/dt to dw/dt. enter your answers using exact values.

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 eighteen more than the width. write an equation that relates dp/dt to dw/dt. enter your answers using exact values.

Answer

Explanation:

Step1: Write the length - width relationship

Given $l = 3w+18$.

Step2: Write the perimeter formula

The perimeter of a rectangle is $P = 2(l + w)$. Substitute $l = 3w + 18$ into it, we get $P=2((3w + 18)+w)=2(4w + 18)=8w+36$.

Step3: Differentiate both sides with respect to $t$

Using the chain - rule, $\frac{dP}{dt}=\frac{d}{dt}(8w + 36)$. Since $\frac{d}{dt}(8w+36)=8\frac{dw}{dt}+0$, we have $\frac{dP}{dt}=8\frac{dw}{dt}$.

Answer:

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