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A rectangular swimming pool is to be build with an area of 150 square meters The owner wants 2-meter wide decks along either side 3m wide decks at the two ends Whats the smallest property?
Wiki User
∙ 12y ago
Supposing that the actual pool is (a)m long and (b)m wide, then the total property will be (a+6)m long and (b+4)m wide.
The next step is to find a relationship between a and b.
ab = 150
Therefore a = 150/b
Therefore, total property area will be:
(150/b + 6)(b+4)
= 150 + 600/b + 6b + 24
= 6b + 600/b + 174
= 6b + 600b-1 + 174
Taking the derivative and setting it to zero:
0 = (6b + 600b-1 + 174)'
= 6 + -1*600*b-2
= 6 - 600/b2
Therefore:
6 = 600/b2
1 = 100/b2
b2 = 100
b = 10
a = 150/10
= 15
To check whether we have the minimum for area, take the second derivative:
(6 - 600/b2)' = (6 - 600b-2)'
= 0 - -2*600b-3
= 1200b-3
= 1200/b3
If b is positive, the second derivative will also be positive, so we do have a minimum.
Therefore total area of the property:
(a+6)m * (b+4)m = (15+6)m * (10+4)m
= 21m * 14m
= 294m2
Wiki User
∙ 12y ago
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