Respuesta :
Answer:
75 square feet
Step-by-step explanation:
The aquarium is designed to hold 62.5cubic feet
Since the aquarium has a square base, assume the aquaruim is a square prism
Let X be the lenght
Let y be the height
x^2.y = 62.5
y = 62.5x^2
The area of each side of the aquarium is xy. Since there are four sides, the total area of the side= 4xy
The area of the bottom of the aquarium = x^2
Total surface area = 4xy + x^2
Put y = 62.5/x^2 into the total surface area equation
A = 4x(62.5/x^2) + x^2
A= 250/x + x^2
Differentiate A with respect to x
dA/dx = -250(x^-2) + 2x
dA/dx = 0
0 = 2x - 250x^-2
2x = 250x^-2
2x =250/ x^2
2x^3 = 250
x^3 = 250/2
x^3 = 125
x = cbrt (125)
x = 5
Put x= 5 into A = x^2 + 250x^-1
A= 5^2 + 250(5^-1)
A= 25 + 250/5
A = 25 + 50
A = 75 square feet
The surface area of a solid object can be determine by measuring the total area that surface of the object cover.
The minimum possible exterior surface area of the aquarium is [tex]75 \:\rm feet^2[/tex].
Given:
The aquarium designed hold [tex]62.5 \:\rm feet^3[/tex].
Let assume the aquarium is square prism.
Let [tex]x[/tex] is height and [tex]y[/tex] is length.
The aquarium designed hold [tex]62.5 \:\rm feet^3[/tex]. Write the system equation.
[tex]x^2y=62.5\\y=\frac{62.5}{x^2}[/tex]
The area of four side of aquarium is as follows,
[tex]\rm A_1=4xy[/tex]
The bottom area of aquarium is as follows,
[tex]A_2=x^2[/tex]
The total area is as follows,
[tex]A=A_1+A_2\\A=4xy+x^2[/tex]
Put [tex]y=\frac{62.5}{x^2}[/tex] in above expression.
[tex]A=4x(\frac{65.2}{x^2} )+x^2\\A=\frac{250}{x}+x^2[/tex]
Differentiate [tex]A[/tex] with respect to [tex]x[/tex].
[tex]\frac{dA}{dx}=-250(x^{-2})+2x[/tex]
For the minimum possible exterior surface area [tex]\frac{dA}{dx} =0[/tex].
[tex]0=2x-250x^{-2}\\2x=250x^{-2}\\2x^3=250\\x^3=\frac{250}{2}\\x^3=125\\x=5[/tex]
Substitute [tex]x=5[/tex] to find the area.
[tex]A=5^2+250\times 5^{-1}\\A=25+\frac{250}{5}\\A=75\:\rm feet^2[/tex]
Thus, the minimum possible exterior surface area of the aquarium is [tex]75 \:\rm feet^2[/tex].
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