Sizing a Cylindrical Hot-Water Tank and Evaluating a Savings Plan
A small brewery is ordering a closed cylindrical hot-water tank with a volume of exactly 2.00 m³. The side wall is made from sheet stainless steel costing 9,000 yen per square metre; the top and bottom disks need extra reinforcement and cost 14,000 yen per square metre. The workshop's rolling machine can produce radii only between 0.40 m and 0.90 m, and the tank must be a right circular cylinder with no overlap allowance. An apprentice argues that the cheapest tank must use the least metal, so the radius should be as small as the machine allows, 0.40 m. A spreadsheet built by another student gives 0.60 m as the minimum. The brewery also wants to know what a maintenance saving of 120,000 yen per year, deposited at the end of each year and earning 2.5% interest compounded annually, would be worth after 8 years. You are asked to settle the radius question rigorously, respecting the machine limit, and to evaluate the savings plan.
1. Write the cost of the tank as a function of radius r, find the radius that minimises cost subject to the machine's limits, and give the minimum cost.
Read the worked answer
Let r be the radius and h the height. The volume constraint gives πr²h = 2.00, so h = 2/(πr²).
Side area = 2πrh = 2πr · 2/(πr²) = 4/r. Top plus bottom area = 2πr².
Cost: C(r) = 9,000 × (4/r) + 14,000 × (2πr²) = 36,000/r + 28,000πr².
Differentiate: C'(r) = −36,000/r² + 56,000πr. Setting C'(r) = 0 gives r³ = 36,000/(56,000π) = 0.2046, so r ≈ 0.589 m. The second derivative, C''(r) = 72,000/r³ + 56,000π, is positive for all r > 0, so the curve is convex and this stationary point is the global minimum.
The value r ≈ 0.589 m lies inside the allowed interval [0.40, 0.90], so no endpoint dominates. Checking anyway: r = 0.40: C = 90,000 + 28,000π(0.16) ≈ 104,100 yen. r = 0.589: C = 36,000/0.589 + 28,000π(0.347) ≈ 61,100 + 30,500 ≈ 91,600 yen. r = 0.90: C = 40,000 + 28,000π(0.81) ≈ 111,250 yen.
The cheapest tank has r ≈ 0.59 m, h = 2/(π × 0.347) ≈ 1.83 m, and costs about 91,600 yen. Because the curve is very flat near the minimum, a radius of 0.60 m costs only about 100 yen more, so the spreadsheet's answer is numerically almost right but not derived from the cost function.
2. The apprentice says 'least metal means least cost'. Show what radius minimises metal area, and explain why that is not the same as minimising cost. Which student is closer to correct, and for what reason?
Read the worked answer
Metal area is A(r) = 4/r + 2πr². Then A'(r) = −4/r² + 4πr, and A' = 0 gives r³ = 1/π, so r ≈ 0.683 m. That is the least-metal radius.
The two answers differ because the two surfaces have different prices: the side wall costs 9,000 yen/m² while the reinforced disks cost 14,000 yen/m². A taller, narrower tank has more side area but smaller disks, and the side material is the cheaper one per square metre, so cost minimisation pushes the design toward a smaller radius than area minimisation does. Equivalently, in C(r) = 36,000/r + 28,000πr² the coefficient of the 1/r term is relatively cheaper than in A(r).
The apprentice is wrong on the reasoning and on the number: clamping r to 0.40 m misses the true minimum and costs roughly 104,100 yen, about 13% more than necessary. The 0.60 m student has the better number but no valid argument. The correct justification is that the optimum of a cost function depends on relative prices, not on total material quantity, and it must be checked against the manufacturing interval.
3. Compute the value after 8 years of depositing 120,000 yen at the end of each year at 2.5% compounded annually, and state the assumption that matters most.
Read the worked answer
This is an ordinary annuity (deposits at year-end), so the last deposit earns no interest: FV = 120,000 × [(1.025⁸ − 1)/0.025].
1.025² = 1.050625; 1.025⁴ = 1.103813; 1.025⁸ = 1.218403. (1.218403 − 1)/0.025 = 0.218403/0.025 = 8.7361. FV = 120,000 × 8.7361 ≈ 1,048,300 yen.
So the plan is worth about 1.05 million yen, of which 960,000 yen is deposits and about 88,000 yen is interest.
The critical assumption is deposit timing. If the 120,000 yen were deposited at the start of each year, every deposit would earn one extra year of interest and the total would be 1.025 times larger (about 1,074,500 yen). The interest rate must also stay fixed at 2.5% and be compounded annually, not monthly. Rounding 1.025⁸ early, or using simple interest (960,000 + 88,000 computed incorrectly as 960,000 × 0.025 × 8), gives wrong totals, so the compounding structure has to be respected.