The length of a rectangular pool is to be four times its width, and a sidewalk 6 feet wide will surround

the pool. If a total area of 1440 𝑓𝑑
2 has been set aside for construction, what are the dimensions of the
pool?

To solve this problem, we can follow these steps:

Step 1: Assign variables to the unknowns.
Let's assume that the width of the rectangular pool is π‘₯ feet. Since the length is four times the width, the length would be 4π‘₯ feet.

Step 2: Calculate the area of the pool.
The area of a rectangle is given by the formula: π΄π‘Ÿπ‘’π‘Ž = π‘™π‘’π‘›π‘”π‘‘β„Ž Γ— π‘€π‘–π‘‘π‘‘β„Ž
In this case, the area of the pool is 𝐴 = (4π‘₯) Γ— π‘₯
Simplifying this, we have 𝐴 = 4π‘₯^2

Step 3: Calculate the area of the entire construction (pool + sidewalk).
The area of the entire construction is given as 1440 𝑓𝑑^2.

Step 4: Set up and solve the equation.
The area of the entire construction is the area of the pool plus the area of the sidewalk.
Thus, we can set up the equation: 1440 = 𝐴 + 𝑆𝑀, where 𝑆𝑀 is the area of the sidewalk.

Step 5: Calculate the area of the sidewalk.
The sidewalk is 6 feet wide and surrounds the pool. Therefore, its width would be the width of the pool plus twice the width of the sidewalk. The width of the sidewalk is given as 6 feet.
So, the width of the sidewalk (π‘Šπ‘ ) can be expressed as π‘Šπ‘  = π‘₯ + 2(6).

The length of the sidewalk is the length of the pool plus twice the width of the sidewalk. The length of the pool is given as 4π‘₯.
So, the length of the sidewalk (𝐿𝑠) can be expressed as 𝐿𝑠 = 4π‘₯ + 2(6).

The area of the sidewalk (𝑆𝑀) can be calculated by multiplying the width and length of the sidewalk.
So, 𝑆𝑀 = π‘Šπ‘  Γ— 𝐿𝑠 = (π‘₯ + 2(6)) Γ— (4π‘₯ + 2(6))

Step 6: Substitute the area of the pool and the area of the sidewalk into the equation.
We can replace 𝐴 and 𝑆𝑀 in the equation 1440 = 𝐴 + 𝑆𝑀 with 4π‘₯^2 and (π‘₯ + 2(6)) Γ— (4π‘₯ + 2(6)) respectively.

1440 = 4π‘₯^2 + (π‘₯ + 2(6)) Γ— (4π‘₯ + 2(6))

Step 7: Solve the equation.
Now, we can simplify and solve the equation to find the value of π‘₯.

With the value of π‘₯, we can determine the dimensions of the pool by using the length and width formulas:
Width of pool = π‘₯
Length of pool = 4π‘₯