Question

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 $\frac{dp}{dt}$ to $\frac{dw}{dt}$. enter your answer using exact values. provide your answer below: $\frac{dp}{dt}=square\frac{dw}{dt}$

Explanation:

Step1: Write perimeter formula

The perimeter formula of a rectangle is $P = 2(l + w)$.

Step2: Substitute the relationship of $l$ and $w$

Given $l=3w + 22$, so $P = 2((3w + 22)+w)=2(4w + 22)=8w + 44$.

Step3: Differentiate both sides with respect to $t$

Using the chain - rule, $\frac{dP}{dt}=\frac{d}{dt}(8w + 44)$. Since $\frac{d}{dt}(8w + 44)=8\frac{dw}{dt}+0$, we can solve for $\frac{dP}{dt}/\frac{dw}{dt}$.
Dividing both sides of $\frac{dP}{dt}=8\frac{dw}{dt}$ by $\frac{dw}{dt}$ (assuming $\frac{dw}{dt}
eq0$), we get $\frac{\frac{dP}{dt}}{\frac{dw}{dt}} = 8$.

Answer:

$8$