question suppose that the length l, width w, and area a = lw of a rectangle are differentiable functions of…

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 da/dt to dl/dt and dw/dt when l = 8 and w = 4. enter your answers using exact values.

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 da/dt to dl/dt and dw/dt when l = 8 and w = 4. enter your answers using exact values.

Answer

Explanation:

Step1: Differentiate area formula

We know $A = lw$. Using the product - rule for differentiation $\frac{d(uv)}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$, where $u = l$ and $v = w$. So $\frac{dA}{dt}=l\frac{dw}{dt}+w\frac{dl}{dt}$.

Step2: Substitute given values

Given $l = 8$ and $w = 4$. Substituting these values into the equation from Step 1, we get $\frac{dA}{dt}=8\frac{dw}{dt}+4\frac{dl}{dt}$.

Answer:

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