The dimensions that minimize the cost of building the tank are r ≈ 6.5 ft and h ≈ 26 ft
What do you mean by cost?
- The cost is the sum of money that a business spends on the development or manufacturing of goods or services. It excludes the profit margin markup.
- The sum of money spent to create a good or product, as seen from the seller's perspective, is the cost.
The total cost becomes
T = ucSc + 2ucSh
= 2ucπrh + 8ucπr²
We need to eliminate h . The volume from the question gives a way out.
4640 = 4/3πr³ + πr²h
h = 4640/πr² - 4r/3
substitute into the formula for total cost gives, after simplifying
T = 16ucπr²/3 + 9280uc/r²
differentiating with respect to r , we get
dT/dr = 32/3Ucπr - 9280Uc/r²
at extrema
dT/dr = 0
⇒ 32/3Ucπr = 9280Uc/r²
r = ∛870/π ≈ 6.5 ft
To confirm that r is a minimum value, carry out the second derivative test
d²T/dr² = 32Ucπ/3 + 18560/r³
substituting ∛870/π, we get that d²T/dr² > 0 confirming that minimum value
To find h , recall that
h = 4640/πr² - 4r/3
substituting r , we get h ≈ 26.0 ft as the corresponding minimum height.
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