Question 177390
The YMCA has a 40ft by 60ft area in which to build a swimming pool. The pool will have a sidewalk of uniform width around 3 sides. (The director of the YMCA does not want a sidewalk along one of the long sides of the pool) Let "x" be the width of the sidewalk.
:
A.) write a quadratic function in terms of "x" expressing the area of the pool
:
draw a  rough diagram if this, one rectangle inside another, showing the walkway
on just 3 sides, a width of x. it will be apparent that the pool dimensions will be:
:
(60-2x) by (40-x), FOIL this and you have: 2400 - 60x - 80x + 2x^2, which is:
:
A(x) = 2x^2 - 140x + 2400; area of the pool
:
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B.) The YMCA would like to construct the pool so that the area of the pool is at least half of the total area set aside for this project. Then state a reasonable interval for "x" that would satisfy this condition
:
Half of the total area = 1200 sq/ft; find the dimensions for this
pool area = 1200
2x^2 - 140x + 2400 = 1200
2x^2 - 140x + 2400 - 1200 = 0
2x^2 - 140x + 1200 = 0
simplify divide by 2;
x^2 - 70x + 600 = 0
Factor
(x-60)(x-10) = 0
two solutions
x = 60
and
x = 10 ft, the only solution that makes sense
We can say the walkway has to be 10 ft wide or less to ensure at least a 1200sq/ft pool
:
:
Check area if walkway is 10' wide:
(60-20) * (40-10) = 1200sq/ft