A rectangular pond 6 meters by 5 meters is going to be built.

Question

A rectangular pond 6 meters by 5 meters is going to be built. The builders want the pond to be surrounded by a walkway 𝑥 meters wide.

A.) Let (𝑥) represent the width and 𝑙(𝑥) represent the length of the region taken up by the pond and the walkway. Write (𝑥) and 𝑙(𝑥) as functions of 𝑥. What kind of functions are they?

B.) Let (𝑥) represent the total area of the region (pond and walkway). Write the area as a function of 𝑥. What kind of function is it?

C.) Calculate the value of 𝑥 that makes the region have an area of 120 square meters.

D.) Calculate the value of 𝑥 that makes the walkway have an area equal to the area of the pond. 

Answer 1

Area = f(x) = (6+2x)(5+2x) = 4x^2 + 22x + 30 square meters

f(x) is a quadratic function, 2 degree polynomial

f(x) = 120 = 4x^2 + 22x +30

4x^2 + 22x -90 = 0

x= -22/8 + or - (1/8)sqr(22^2 +16(90))

x = -11/4 + or - (1/8)sqr(1924) = about 2.5 meters (ignore the negative square root) for area=120

Area of the walkway is 10x+12x + 4x^2 = area of pond = 6(5) = 30

4x^2 +22x -30 =0

x = -22/8 + or - (1/8)sqr(22^4(4)(30)) = -11/8 + (1/8)sqr964= about 1, rounded off to nearest integer

But it's slightly larger than 1, as 22(1)+4(1) = just 26 for the walkway area < 30 for the pond

It's slightly bigger than x=1.1 meters 22(1.1)+4(1.1)^2 = 29.04 square meters