Answer:
A. The sidewalk length: 24.9'
B. The sidewalk width: 21.6'
C. The garden length: 11.7'
D. The garden width: 8.4'
E. Total garden area: 98.3 square inches
Explanation:
We have two rectangles (R). Let's define:
R₁ = the biggest rectangle = sidewalk + garden
R₂ = the smallest rectangle = garden
The area of the rectangle R₁ (AR₁) can be calculated as follows:
AR₁ = l₁.w₁
AR₁ = (1.87x+5+x+3+x+3)(1.5x+3+x+3+x+3)
AR₁ = (3.87x + 11)(3.5x + 9)
Also,
AR₁ = area of sidewalk + area of garden
Area of the sidewalk = 64875 square inches = 450.5 square feet
AR₁ = 450.5 + (1.87x + 5)(1.5x + 3)
We can equal both equations to find the value of x:
(3.87x + 11)(3.5x + 9) = 450.5 + (1.87x + 5)(1.5x + 3)
13.5x² + 34.8x + 38.5x + 99 = 450.5 + (2.8x² + 5.6x + 7.5x + 15)
13.5x² + 73.3x + 99 = 2.8x² + 13.1x + 450.5
13.5x² -2.8 x² + 73.3x - 13.1x + 99 - 450.5 = 0
10.7x² + 60.2x - 351.5 = 0
Now, we can use the quadratic formula to the value of x.
According to the quadratic formula:
For a equation:
ax²+ bx + c = 0,
x = (-b ± √Δ)/2a
and
Δ = b² - 4ac
So, in this exercise:
Δ = 60.2² - 4(10.7)(-351.5)
Δ = 3624 + 15044.2
Δ = 18668.2
x = (-60.2 ± √18668.2)/(2*10.7)
x = (-60.2 ± 136.6)/21.4
x₁ = (-60.2 + 136.3)/21.4
x₁ = 3.6'
x₂ = (-60.2 - 136.3)/21.4
x₂ = -9.2'
Since the sides of the rectangle can not be negative, we will use the value of x₁ = 3.6'.
Now, let's calculate the sides of the garden:
lenght: 1.87x + 5
length: 1.87*3.6 + 5
length: 11.7'
width: 1.5x + 3
width: 1.5*3.6 + 3
width: 8.4'
And the area of the garden AR₂:
AR₂ = 11.7*8.4
AR₂ = 98.3 square inches
Finally, let's calculate the sides of the biggest rectangle:
lenght: 11.7 + x + 3 + x + 3
lenght: 11.7 + 3.6 + 3 + 3.6 +3
lenght: 24.9'
width: 8.4 + x + 3 x + 3
width: 8.4 + 3.6 + 3 + 3.6 +3
width: 21.6'