A company is constructing an open-top, square based, rectangular metal tank that will have a volume of 27ft^3. What dimensions yield the minimum surface area? Round to the nearest tenth, if necessary.
A) 4.3ft x 4.3ft . 1.4 ft
B) 7.3ft x 7.3ft x 0.5ft
C) 3.8ft x 3.8ft x 1.9ft
D) 3ft x 3ft x 3ft
Please explain!! :)
A) 4.3ft x 4.3ft . 1.4 ft
B) 7.3ft x 7.3ft x 0.5ft
C) 3.8ft x 3.8ft x 1.9ft
D) 3ft x 3ft x 3ft
Please explain!! :)
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If 'x' represents a side length in the base and 'h' represents height, V = x^2*h = 27
h = 27/x^2
Surface Area = A(x) = x^2 + 4xh (bottom base and 4 sides)
= x^2 + 4(x)(27/x^2)
= x^2 + 108/x
Differentiate the surface area function and set it equal to zero (this is your minimum).
A'(x) = 2x - 108/x^2 = 0
2x = 108/x^2
2x^3 = 108
x^3 = 54
x = cube root(54)
x = about 3.8 ft.
Find the height now.
h = 27/3.8^2 ft = about 1.9 ft.
Dimensions = 3.8 ft by 3.8 ft by 1.9 ft
Answer: c)
h = 27/x^2
Surface Area = A(x) = x^2 + 4xh (bottom base and 4 sides)
= x^2 + 4(x)(27/x^2)
= x^2 + 108/x
Differentiate the surface area function and set it equal to zero (this is your minimum).
A'(x) = 2x - 108/x^2 = 0
2x = 108/x^2
2x^3 = 108
x^3 = 54
x = cube root(54)
x = about 3.8 ft.
Find the height now.
h = 27/3.8^2 ft = about 1.9 ft.
Dimensions = 3.8 ft by 3.8 ft by 1.9 ft
Answer: c)
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D) the minimal surface area is 3ft x 3ft