Question

question suppose that the length l, width w, and area a = lw of a rectangle are differentiable functions of t. write an equation that relates $\frac{da}{dt}$ to $\frac{dl}{dt}$ and $\frac{dw}{dt}$ when l = 8 and w = 4. enter your answers using exact values. provide your answer below: $\frac{da}{dt}=square\frac{dl}{dt}+square\frac{dw}{dt}$

Explanation:

Step1: Differentiate area formula

The area of a rectangle is $A = lw$. Differentiate both sides with respect to $t$ using the product - rule $(uv)^\prime=u^\prime v + uv^\prime$, where $u = l$ and $v = w$.
By the product - rule, $\frac{dA}{dt}=\frac{dl}{dt}w + l\frac{dw}{dt}$.

Step2: Substitute given values

We are given $l = 8$ and $w = 4$. Substituting these values into the derivative formula, we get $\frac{dA}{dt}=4\frac{dl}{dt}+8\frac{dw}{dt}$.

Answer:

$\frac{dA}{dt}=4\frac{dl}{dt}+8\frac{dw}{dt}$