Answer:
The width and the length of the pool are 12 ft and 24 ft respectively.
Step-by-step explanation:
The length (L) of the rectangular swimming pool is twice its wide (W):
[tex] L_{1} = 2W_{1} [/tex]
Also, the area of the walkway of 2 feet wide is 448:
[tex] W_{2} = 2 ft [/tex]
[tex] A_{T} = W_{2}*L_{2} = 448 ft^{2} [/tex]
Where 1 is for the swimming pool (lower rectangle) and 2 is for the walkway more the pool (bigger rectangle).
The total area is related to the pool area and the walkway area as follows:
[tex] A_{T} = A_{1} + A_{w} [/tex] (1)
The area of the pool is given by:
[tex] A_{1} = L_{1}*W_{1} [/tex]
[tex] A_{1} = (2W_{1})*W_{1} = 2W_{1}^{2} [/tex] (2)
And the area of the walkway is:
[tex] A_{w} = 2(L_{2}*2 + W_{1}*2) = 4L_{2} + 4W_{1} [/tex] (3)
Where the length of the bigger rectangle is related to the lower rectangle as follows:
[tex] L_{2} = 4 + L_{1} = 4 + 2W_{1} [/tex] (4)
By entering equations (4), (3), and (2) into equation (1) we have:
[tex] A_{T} = A_{1} + A_{w} [/tex]
[tex]A_{T} = 2W_{1}^{2} + 4L_{2} + 4W_{1}[/tex]
[tex]448 = 2W_{1}^{2} + 4(4 + 2W_{1}) + 4W_{1}[/tex]
[tex] 224 = W_{1}^{2} + 8 + 4W_{1} + 2W_{1} [/tex]
[tex] 224 = W_{1}^{2} + 8 + 6W_{1} [/tex]
By solving the above quadratic equation we have:
W₁ = 12 ft
Hence, the width of the pool is 12 feet, and the length is:
[tex] L_{1} = 2W_{1} = 2*12 ft = 24 ft [/tex]
Therefore, the width and the length of the pool are 12 ft and 24 ft respectively.
I hope it helps you!
