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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.

Respuesta :

Answer:

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

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