A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a 2 feet wide and ha

Question

A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a 2 feet wide and has an area of 448 square feet including the pool.Find the dimensions of the pool

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Hồng Cúc 2 weeks 2021-08-30T19:01:39+00:00 1 Answers 0 views 0

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    2021-08-30T19:02:41+00:00

    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):

     L_{1} = 2W_{1}

    Also, the area of the walkway of 2 feet wide is 448:

     W_{2} = 2 ft

     A_{T} = W_{2}*L_{2} = 448 ft^{2}

    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:

     A_{T} = A_{1} + A_{w}    (1)          

    The area of the pool is given by:

     A_{1} = L_{1}*W_{1}        

     A_{1} = (2W_{1})*W_{1} = 2W_{1}^{2}  (2)          

    And the area of the walkway is:

     A_{w} = 2(L_{2}*2 + W_{1}*2) = 4L_{2} + 4W_{1}    (3)          

    Where the length of the bigger rectangle is related to the lower rectangle as follows:                  

     L_{2} = 4 + L_{1} = 4 + 2W_{1}   (4)        

    By entering equations (4), (3), and (2) into equation (1) we have:

     A_{T} = A_{1} + A_{w}

    A_{T} = 2W_{1}^{2} + 4L_{2} + 4W_{1}                

    448 = 2W_{1}^{2} + 4(4 + 2W_{1}) + 4W_{1}            

     224 = W_{1}^{2} + 8 + 4W_{1} + 2W_{1}

     224 = W_{1}^{2} + 8 + 6W_{1}

    By solving the above quadratic equation we have:

    W₁ = 12 ft

    Hence, the width of the pool is 12 feet, and the length is:

     L_{1} = 2W_{1} = 2*12 ft = 24 ft

    Therefore, the width and the length of the pool are 12 ft and 24 ft respectively.

    I hope it helps you!                                                                                          

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