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

129
`cm^2/s`

Explanation:

Increasing rate of length,
`(dl)/(dt)`= 9 cm/s

Increasing rate of width,
`(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 * Width`

OR

`A = l * w`

Differentiating w.r.to t:

`(d)/(dt)A = (d)/(dt)(l * w)\\\Rightarrow (d)/(dt)A = w * (d)/(dt)l +l * (d)/(dt)w`

Putting the values:

`\Rightarrow (dA)/(dt) = 5 * 9 + 12 * 7\\\Rightarrow (dA)/(dt) = 45 + 84\\\Rightarrow \bold{(dA)/(dt) = 129\ cm^2/sec}`