A city is in the shape of a rectangle. In 1995 the width of the city was 9 miles and the length of the city was 5 miles. The width of the city is growing at a rate of 1 mile in 9 years. The length of the city is growing at a rate of 1 mile in 6 years. Use the product rule to find how quickly the area of the city is growing in 1995.

DA/dt = 37 / 18

Step-by-step explanation:

We have the following information:

Sides of the rectangle L = length w = width

Area of the rectangle is : A = L * w

the rate of growing of the length (as function of time) 1 mile in 6 years

we can express that as:

DL/dt = 1 / 6

And the rate of growing of the width (as function of time) 1 mile in 9 years

Dw/dt = 1/9

In 1995 dimensions of the rectangle were

L = 5 miles and w = 9 miles

Then:

A = L * w Taking derivatives

DA/dt = L * Dw/dt + w * DL/dt

DA/dt = 5 * (1/9) + 9 * (1/6) ⇒ DA/dt = 5/9 + 9/6

DA/dt = 111 / 54 ⇒ DA/dt = 37 / 18