Q58. - Calculus

Question

Suppose the width w of a rectangle is decreasing at a

rate of 3 inches per second while the height h of the rectangle

is increasing at a rate of 3 inches per second. The

rectangle initially has a width of 100 inches and a height

of 75 inches.

(a) Find the rate of change of the area of the rectangle in

terms of its width and height.

(b) On what intervals do the variables w and h make

sense in this problem? On what time interval does

the problem make sense?

(c) When will the area of the rectangle be increasing, and

when will it be decreasing? Answer these questions

both in terms of the width and height of the rectangle

and in terms of time.

Step-by-Step Solution

Answer

(a) Rate of change of area is$=-3h+3w$

(b) The problem will make sense for $h=325, w=0 \& t=\frac{100}{3}$.

(c) The area of the rectangle is increasing when $w>0$ and decreasing when  $w=0.$

Part (a) Step 1. Given Information

The rate of decreasing the width of the rectangle is $3.$ 

The rate of increase in height of the rectangle is $3.$

The initial height and width of the rectangle  inches $75$and $100$ inches.

Part (a) Step 2. Calculation

Here, $\frac{dw}{dt}=-3 \& \frac{dh}{dt}=3$

Now the area of a rectangle is

$A=hw$

Differentiating both sides with respect to $"t"$we get

$\frac{dA}{dt}=h.\frac{dw}{dt}+w.\frac{dh}{dt}$

$=-3h+3w$

Part (b) Step 1. Given Information

The rate of decreasing the width of the rectangle is $3.$

The rate of increase in height of the rectangle is $3.$

The initial height and width of the rectangle is $75$ inches and $100$ inches.

Part (b) Step 2. Calculation

We have, $\frac{dw}{dt}=-3 \& \frac{dh}{dt}=3$

Now integrating the above two equations we get

$w=-3t+c \& h=3t+k$

At $t=0, w=100 \& h=75$

Then $100=c \& 25=k$

Therefore, $w=-3t+100 \& h=3t+25$

Then the problem will make sense if $w=0$$w=0$

i.e. if $t=\frac{100}{3}$sec.

Part (c) Step 1. Given Information

The rate of decreasing the width of the rectangle is $3.$

The rate of increase in height of the rectangle is $3.$

The initial height and width of the rectangle is $75$inches and$100$ inches.

Part (c) Step 2. Explanation

The area of the rectangle increases when $w>0$and decreases when $w\le 0.$