Question

The length of a rectangle is increasing at a rate of 4 cm/s and its width is increasing at a rate of 4 cm/s. When the length is 5 cm and the width is 4 cm, how fast is the area of the rectangle increasing

Answer 1

Let us set up the following variables:

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

l

Length of Rectangle (cm)

w

Width of Rectangle (cm)

A

Area of Rectangle (

c

m

2

)

t

Time (s)

We are told that:

d

l

d

t

=

8

cm/s (const), and,

d

w

d

t

=

3

cm/s (const)

The Area of the rectangle is:

A

=

l

w

Differentiating wrt

t

(using the product rule) we get;

d

A

d

t

=

(

l

)

(

d

w

d

t

)

+

(

d

l

d

t

)

(

w

)

∴

d

A

d

t

=

3

l

+

8

w

So when

l

=

20

and

w

=

10

⇒

d

A

d

t

=

3

⋅

20

+

8

⋅

10

∴

d

A

d

t

=

60

+

80

∴

d

A

d

t

=

140

cm

2

/

s