Answer:
Length and Width = 10ft
Height = 5ft
Surface Area = 300 square feet
Step-by-step explanation:
Given
[tex]V = 500ft^3[/tex] -- Volume
Let:
[tex]L = Length[/tex]
[tex]W =Width[/tex]
[tex]H = Height[/tex]
Volume (V) is calculated as:
[tex]V = L * W * H[/tex]
Substitute 500 for V
[tex]500 = L * W * H[/tex]
Make H the subject
[tex]H = \frac{500}{LW}[/tex]
The tank has no top. So, the surface area (S) is:
[tex]S = L * W + 2*H*L + 2*H*W[/tex]
[tex]S = L * W + 2H(L + W)[/tex]
Substitute 500/LW for H
[tex]S = L * W + 2*\frac{500}{LW}(L + W)[/tex]
[tex]S = L * W + \frac{1000}{LW}(L + W)[/tex]
[tex]S = L W + \frac{1000}{L} + \frac{1000}{W}[/tex]
Differentiate with respect to L and to W
[tex]S'(W) = L - \frac{1000}{W^2}[/tex]
and
[tex]S'(L) = W - \frac{1000}{L^2}[/tex]
Equate both to get the critical value
[tex]S'(W) = L - \frac{1000}{W^2}[/tex]and [tex]S'(L) = W - \frac{1000}{L^2}[/tex]
[tex]0 = L - \frac{1000}{W^2}[/tex] and [tex]0 = W - \frac{1000}{L^2}[/tex]
[tex]\frac{1000}{W^2} = L[/tex] and [tex]\frac{1000}{L^2} = W[/tex]
[tex]W^2L = 1000[/tex] and [tex]L^2W = 1000[/tex]
Make L the subject in [tex]W^2L = 1000[/tex]
[tex]L = \frac{1000}{W^2}[/tex]
Substitute [tex]\frac{1000}{W^2}[/tex] for L in [tex]L^2W = 1000[/tex]
[tex](\frac{1000}{W^2})^2 * W = 1000[/tex]
[tex]\frac{1000000}{W^4} * W = 1000[/tex]
[tex]\frac{1000000}{W^3} = 1000[/tex]
Cross Multiply
[tex]1000000 = 1000W^3[/tex]
Divide both sides by 1000
[tex]1000 = W^3[/tex]
Take cube roots of both sides
[tex]\sqrt[3]{1000} = W[/tex]
[tex]10 = W[/tex]
[tex]W = 10[/tex]
Substitute 10 for W in [tex]L = \frac{1000}{W^2}[/tex]
[tex]L = \frac{1000}{10^2}[/tex]
[tex]L = \frac{1000}{100}[/tex]
[tex]L = 10[/tex]
Recall that:[tex]H = \frac{500}{LW}[/tex]
[tex]H = \frac{500}{10*10}[/tex]
[tex]H = \frac{500}{100}[/tex]
[tex]H = 5[/tex]
So, the dimensions are:
[tex]L, W=10[/tex] and [tex]H = 5[/tex]
The surface area is:
[tex]S = L * W + 2H(L + W)[/tex]
[tex]S = 10*10 +2*5(10+10)[/tex]
[tex]S = 10*10 +2*5*20[/tex]
[tex]S = 100 + 200[/tex]
[tex]S = 300[/tex]
