Question

step 1
we need to determine how fast the area of the rectangle is increasing. in other words, we are looking for a rate of change of the area. in this problem, the volume, the length, and the width are all functions of the time t, where t is measured in seconds.
the rate of increase of the area with respect to time is the derivative \\(\frac{da}{dt}\\). the rate of increase of the length and width with respect to time are the derivatives \\(\frac{dl}{dt}\\) and \\(\frac{dw}{dt}\\) respectively.
therefore, we need to differentiate each side of our area - equation with respect to t using the product - rule. doing so gives the following result where \\(\frac{da}{dt}\\) is measured in \\(cm^{2}/s\\).
\\(\frac{da}{dt}=\frac{d(lw)}{dt}\\)
\\(\frac{da}{dt}=(\square)\frac{dw}{dt}+(\square)\frac{dl}{dt}\\)
submit
skip (you cannot come back)
resources

Explanation:

Step1: Recall product - rule for differentiation

The product - rule states that if $y = uv$, where $u$ and $v$ are functions of $t$, then $\frac{dy}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$. In the case of the area of a rectangle $A = lw$ (where $l$ is the length and $w$ is the width, both functions of $t$), by the product - rule, $\frac{dA}{dt}=l\frac{dw}{dt}+w\frac{dl}{dt}$.

Answer:

The first blank is $l$, the second blank is $w$.