Question

the length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 2 cm/s. when the length is 30 cm and the width is 10 cm, how fast is the area of the rectangle increasing? answer (in cm²/s):

Explanation:

Step1: Recall the area formula

Let the length of the rectangle be $l$ and the width be $w$. The area $A = lw$.

Step2: Differentiate with respect to time $t$

Using the product - rule $\frac{dA}{dt}=l\frac{dw}{dt}+w\frac{dl}{dt}$.

Step3: Substitute the given values

We know that $\frac{dl}{dt}=8$ cm/s, $\frac{dw}{dt}=2$ cm/s, $l = 30$ cm and $w = 10$ cm.
Substituting into the formula: $\frac{dA}{dt}=30\times2 + 10\times8$.
$\frac{dA}{dt}=60 + 80$.
$\frac{dA}{dt}=140$ cm²/s.

Answer:

140