Question

The length of a rectangle is increasing at a rate of 9 cm/s and its width is increasing at a rate of 7 cm/s. When the length is 12 cm and the width is 5 cm, how fast is the area of the rectangle increasing?

Answer 1

Answer:

129 $cm^2/s$

Step-by-step explanation:

Increasing rate of length, $\frac{dl}{dt}$= 9 cm/s

Increasing rate of width, $\frac{dw}{dt}$ = 7 cm/s

Length, l = 12 cm

Width, w = 5 cm

To find:

Rate of increase of area of rectangle at above given points.

Solution:

Formula for area of a rectangle is given as:

$Area = Length \times Width$

OR

$A = l \times w$

Differentiating w.r.to t:

$\dfrac{d}{dt}A = \dfrac{d}{dt}(l \times w)\\\Rightarrow \dfrac{d}{dt}A = w \times \dfrac{d}{dt}l +l \times \dfrac{d}{dt}w$

Putting the values:

$\Rightarrow \dfrac{dA}{dt} = 5 \times 9 + 12 \times 7\\\Rightarrow \dfrac{dA}{dt} = 45 + 84\\\Rightarrow \bold{\dfrac{dA}{dt} = 129\ cm^2/sec}$