Question

related rates: problem 2 (1 point) the length of a rectangle is increasing at a rate of 6cm/s and its width is increasing at a rate of 3cm/s. when the length is 30cm and the width is 10cm, how fast is the area of the rectangle increasing? answer (in cm²/s):

Explanation:

Step1: Write the area formula

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

Step2: Differentiate with respect to time $t$

Using the product - rule $\frac{d(uv)}{dt}=u\frac{dv}{dt}+v\frac{du}{dt}$, we have $\frac{dA}{dt}=l\frac{dw}{dt}+w\frac{dl}{dt}$.

Step3: Substitute the given values

We know that $\frac{dl}{dt}=6$ cm/s, $\frac{dw}{dt}=3$ cm/s, $l = 30$ cm and $w = 10$ cm.
Substituting these values into the formula for $\frac{dA}{dt}$, we get $\frac{dA}{dt}=30\times3 + 10\times6$.
$\frac{dA}{dt}=90 + 60$.

Answer:

$150$