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4 July, 16:40

A pool measuring 8 meters by 26 meters is surrounded by a path of uniform width, as shown in the figure. If the area of the pool and the path combined is 1008 square meters, what is the width of the path?

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  1. 4 July, 16:42
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    Let the width path be x.

    Length of the outer rectangle = 26 + 2x.

    Width of the outer rectangle = 8 + 2x.

    Combined Area = (2x + 26) * (2x + 8) = 1008

    2x * (2x + 8) + 26 * (2x + 8) = 1008

    4x² + 16x + 52x + 208 = 1008

    4x² + 68x + 208 - 1008 = 0

    4x² + 68x - 800 = 0. Divide through by 4.

    x² + 17x - 200 = 0. This is a quadratic equation.

    Multiply first and last coefficients: 1*-200 = - 200

    We look for two numbers that multiply to give - 200, and add to give + 17

    Those two numbers are 25 and - 8.

    Check: 25*-8 = - 200 25 + - 8 = 17

    We replace the middle term of + 17x in the quadratic expression with 25x - 8x

    x² + 17x - 200 = 0

    x² + 25x - 8x - 200 = 0

    x (x + 25) - 8 (x + 25) = 0

    (x+25) (x - 8) = 0

    x + 25 = 0 or x - 8 = 0

    x = 0 - 25 x = 0 + 8

    x = - 25 x = 8

    The width of the path can not be negative.

    The only valid solution is x = 8.

    The width of the path is 8 meters.
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