What must the width of the lawn be?

Question

A factory is to be built on a lot measuring 300 ft by 400ft. A local building code specifies that a lawn of uniform width and equal area to the factory must surround the factory. What must the width of the lawn be?

 

If the dimensions of the factory are A ft by B ft with A<B, then A=? and B=?

Answer 1

Hi Roselyn,

 

Let's begin by representing our unknown in terms of a letter:

    x (width of surrounding lawn in feet)

 

Now let's calculate the area of the factory:

    Area = length x width

    Area = 400 ft. x 300 ft.

    Area = 120000 sq. ft.

 

Since the area of the surrounding lawn must equal the area of the factory then their combined area must be twice the area of the factory:

    Combined Area = 2 x 120000 sq. ft. 

    Combined Area = 240,000 sq. ft.

 

Now let's write an expression for the combined area using the dimensions of the factory and our unknown

    Combined Area = (400 + 2x)(300 + 2x)

 

Note that we add twice the surrounding lawn width to each dimension of the factory, because the surrounding lawn is on both sides of each dimension of the factory.

 

Now let's equate the Combined Area expression to its previously calculated value and solve for our unkown (x):

    (400 + 2x)(300 + 2x) = 240000

    120000 + 800x + 600x + 4x² = 240000

    120000 + 1400x + 4x² = 240000

    4x² + 1400x - 120000 = 0

    4(x² + 1400x - 30000) = 0

    x² + 350x - 30000 = 0

    x² + 350x = 30000

    x² + 350x + (350/2)² = 30000 + (350/2)²

    x² + 350x + (175)² = 30000 + (175)²

    (x + 175)² = 30000 + 30625

    (x + 175)² = 60625

    x + 175 = +/- √60625

    x + 175 = +/- 246.22

             x = -175 - 246.22

             x = -421.22 feet (reject negative

                                      measurement)

             x = -175 + 246.22

             x = 71.22

                   (width of surrounding lawn in feet)

 

The key to solving this problem was that according to the given information, we needed to see that the combined area (factory + lawn) was equal to twice the area of the factory, which allowed us to set up our equation in terms of our unknown.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.