Answer:
7 feet by 7 feet by 3.5 feet
Explanation:
The volume of the tank which is open at the top = 171.5 ft³
Where:
I = length
w = width
h = height
V = Volume
The surface area of the tank because it is open at the top = lw + 2wh + 2lh..... Equation 1
Volume of the tank =lwh....... Equation 2
Hence,
h = V/lw...... Equation 3
Putting this into our Equation 1
lw + 2w(V/lw) + 2l(V/lw)
lw + 2V/l + 2V/w
Finding the derivative
dy/dl = w - 2V/l²
dy/dw = l - 2V/w²
Hence:
w - 2V/l² = 0
w = 2V/l²....... Equation 4
l - 2V/w² = 0
l = 2V/w².......... Equation 5
l = 2V/w².......... Equation 5
Note : w = 2V/l², so we substitute
l = 2V/(2V/l²)²
l = 2V ÷ (4V²/l⁴)²
I = 2V × l⁴/4V²
I = l⁴/2V
Cross Multiply
I × 2V = l⁴
Divide both sides by l
2V = l⁴/l
2V = l³
l³ = 2V
V = 171.5cm³
l = cube root(2 × 171.5)
l = cube root (343)
l = 7 feet
w = 2V/l²....... Equation 4
w = 2 × 171.5/7²
w = 343/49
w = 7 feet
h = V/lw...... Equation 3
h = 171.5/7 × 7
h = 171.5/49
h = 3.5 feet
Therefore, the dimensions of the tank that require the least material to build;
7 feet by 7 feet by 3.5 feet
Where:
7feet = Length
7 feet = Width
3.5 feet = Height