The pool dimensions that will minimize the cost of construction are length of square bottom = 20 ft and height of pool = 5 ft
Let l be the length of side of the square bottom of the pool and h be the height of the pool.
Its volume V = l²h
Now, the total surface area of the pool equal the area of its sides plus the area of its base.
The area of its 4 sides is 4lh.
Since the cost of the side of the pool is $80 per square foot, the total cost for the sides of the pool is C = $80 × 4lh = $ 320lh
Also, The area of its square bottom is l².
Since the cost of the side of the pool is $40 per square foot, the total cost for the square bottom of the pool is C' = $40 × l² = $ 40l²
So, the total cost of the pool is C" = C + C' = 320lh + 40l²
C" = 320lh + 40l²
Since the volume equals 2000 ft³, V = l²h
2000 = l²h
h = 2000/l²
Substituting h into C", we have
C" = 320lh + 40l²
C" = 320l × 2000/l² + 40l²
C" = 640000/l + 40l²
To find the value of l which makes C" minimum, we differentiate C" with respect to l and equate it to zero.
So, dC"/dl = d(640000/l + 40l²)/dl
dC"/dl = d(640000/l)/dl + d(40l²)/dl
dC"/dl = -640000/l² + 80l
Equating dC"/dl to zero, we have
-640000/l² + 80l = 0
-640000/l² = -80l
Cross-multiplying, we have
640000 = 80l³
Dividing through by 80, we have
l³ = 640000/80
l³ = 8000
Taking cube root of both sides, we have
l = ∛8000
l = 20 ft
We differentiate C" twice to determine if this value of l gives a minimum for C".
So, d²C"/dl² = d(-640000/l² + 80l)/dl
d²C"/dl² = d(-640000/l²)/dl + d(80l)/dl
d²C"/dl² = 640000/l³ + 80
Substituting l = 20, we have
d²C"/dl² = 640000/l³ + 80
d²C"/dl² = 640000/20³ + 80
d²C"/dl² = 640000/8000 + 80
d²C"/dl² = 80 + 80
d²C"/dl² = 160 > 0
Since d²C"/dl² = 160 > 0. So, l = 20 ft gives a minimum for C".
Since h = 2000/l²
Substituting l = 20 into the equation, we have
h = 2000/l²
h = 2000/20²
h = 2000/400
h = 5 ft
So, the pool dimensions that will minimize the cost of construction are length of square bottom = 20 ft and height of pool = 5 ft
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