Question

related rates: problem 2 (1 point) the length of a rectangle is increasing at a rate of 8cm/s and its width is increasing at a rate of 6cm/s. when the length is 40cm and the width is 20cm, how fast is the area of the rectangle increasing? answer (in cm²/s):

Explanation:

Step1: Recall the area formula

The area formula of a rectangle is $A = lw$, where $l$ is the length and $w$ is the width.

Step2: Differentiate with respect to time

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 are given that $\frac{dl}{dt}=8$ cm/s, $\frac{dw}{dt}=6$ cm/s, $l = 40$ cm and $w = 20$ cm.
Substitute these values into the formula: $\frac{dA}{dt}=40\times6 + 20\times8$.

Step4: Calculate the result

$\frac{dA}{dt}=240+160=400$ cm²/s.

Answer:

400