Question

A rectangular piece of land to be developed as a memorial park has an area of 75 m^2. The length of the lot is three times the width of the lot. A rectangular path whose width is x meters is to be constructed along the inner perimeter of the lot. The land contained within the landscape the inner field is $3.00 per square meter. express the total cost to develop this lot as a function of x, the width of the path.

Answer 1

Let length = l

width = w

length = 3 times width = 3w

Area of the land = 75 sq. m

Area = length × width = 3w×w = 75

3w²=75

w²=25

w=5 m

length = 3×15 = 15m

Now lets make the figure with the path of width x meters

(Refer the attached figure for easy understanding)

The inner length is the length of the rectangle reduced by x meters on either side due to the path

that is 15-x-x = 15-2x

the inner width is the width reduced by x meters in either side

= 5-x-x = 5-2x

Now lets calculate the inner area = inner length × inner width

(15-2x)×(5-2x)

= $75-30x-10x+4x^{2}$

= $75-40x+4x^{2}$ sq. m

Now the function to calculate the cost

f(x) is given by $3.00 × area of the inner land

f(x) = 3 × {75-40x+4x²)

= 225-120x+12x²

or

= 12x²-120x+225