Question

The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 5 cm/s. When the length is 13 cm and the width is 10 cm, how fast is the area of the rectangle increasing?

Answer 1

Answer:

145 cm²/s

Explanation:

Let the length be 'L', width be 'w' and area be 'A' at any time 't'.

Given:

Rate of increase of length is, $\frac{dL}{dt}=8\ cm/s$

Rate of increase of width is, $\frac{dw}{dt}=5\ cm/s$

Area of the rectangle is given as:

$A=Lw$

Differentiating both sides with respect to time 't', we get;

$\frac{dA}{dt}=\frac{d}{dt}(Lw)\\\\\frac{dA}{dt}=w\frac{dL}{dt}+L\frac{dw}{dt}$

Now, we need to find the rate of area increase when L = 13 cm and w = 10 cm. So, plug in all the given values and solve for $\frac{dA}{dt}$. This gives,

$\frac{dA}{dt}=(10)(8)+(13)(5)\\\\\frac{dA}{dt}=80+65=145\ cm^2/s$

Therefore, the area is increasing a a rate of 145 cm²/s.