Skip to content
Optimisation Problems
🧠 Optimisation Problems: Minimising Surface Area (Water Tank Example)
Okay, let’s get into this — and honestly, before we even start, let me say this: optimisation questions look harmless and then suddenly half the room is frowning 🎯. Happens every year. You think it’s just a cylinder, right? But then the shape starts changing in your head and the numbers push back and, well… it gets interesting pretty fast 📘. When I go through this with students during A Level Maths Revision, there’s usually one person whispering, “Wait, why would we ever make it taller?” and someone else saying the exact opposite. So yes — it’s a good one. And we’ll take it slowly, because rushing this topic is a guaranteed headache.
🔙 Previous topic:
If that surface-area problem made sense but the calculus felt heavier than you wanted, it might be worth stepping back to Integration by Parts: When (and When Not) to Use It just to clean the gears before the next jump.
📘 Exam Context
Here’s the thing examiners love about these: they look innocent but test almost everything — modelling, logic, algebra, differentiation, judgement. And because the question’s wrapped in a real-world style, students sometimes overthink it instead of grounding themselves in the usual steps. Edexcel, AQA, OCR… they all throw this type in. And if you’re doing A Level Maths, by the way, this style is guaranteed to show up somewhere. One more little warning: exam reports repeatedly mention that the “state a minimum” step goes missing. Honestly, I’ve said “Don’t skip the check” more times than I’ve said “Good morning” in June.
📏 Problem Setup
A closed cylindrical water tank must hold 2000\pi cubic metres of water.
The job is to find the radius and height that minimise the total surface area.
Read that twice — it’s surprisingly easy to skim past the important bit.
🖼️ Required Diagram
🧠 Let’s Just Talk Through the Situation First
Right, before we lunge at formulas, let me chat through what’s actually happening. The volume is fixed — that never moves — so if we fiddle with the radius, the height has to shift to keep the volume constant. And honestly, that alone is where I pause in lessons because someone always says, “So could the tank be ridiculously tall if the radius is tiny?” Yes. It absolutely could. (It would look absurd, but mathematically it works.) And that’s why there must be a most efficient combination somewhere in the middle. We’re basically hunting for that sweet spot.
📘 The Volume Formula — Just Enough Detail
We only need one formula to begin:
V = \pi r^2 h.
Nothing wild. Since the volume has already been fixed at 2000\pi, I just turn that around and get the height in terms of the radius.
I’m not dragging you through every algebra shuffle — you’ve done rearranging since Year 9 — but the height ends up as:
h = \frac{2000}{r^2}.
If your instinct says “So height goes up when radius goes down,” yes, exactly — that’s the little tug-of-war we’ll use later.
📏 Surface Area: The Model That Matters
A closed cylinder has:
- one circular top,
- one circular base,
- and the curved surface.
Put together:
S = 2\pi r^2 + 2\pi r h.
This is the line where — honestly — half of my Year 13 group forget the second circle. So double-check: closed tank = two circles.
Now we pop the height expression into that model. Behind the scenes that tidies everything into a single-variable formula. That’s the bit that makes differentiation possible.
⚙️ After Substitution — Ready for the Interesting Part
Once height is replaced, we end up with something tidy:
S(r) = 2\pi r^2 + \frac{4000\pi}{r}.
Don’t let the reciprocal spook you — it’s perfectly normal in modelling.
Differentiation next — and I’ll just say this plainly — slow down when you hit negative powers. I’ve watched top-grade students lose two marks in five seconds because they moved too confidently.
Differentiating gives:
S'(r) = 4\pi r – 4000\pi r^{-2}.
This is the “pause and check your sign” moment.
⚙️ Solving for the Best Radius
Setting the derivative equal to zero pinpoints the moment the surface area stops changing. I’m not writing out the whole rearrangement dance — because it’s mechanical — but trust me, everything collapses nicely into:
r = 10.
Very exam-board-friendly.
Then height drops straight out by substitution:
h = 20.
So: radius 10 m, height 20 m.
🧠 We Still Need to Show It’s a Minimum
Right — don’t pack away your pen yet. This is the part almost everyone forgets. A stationary point is just that: stationary. Could be a min, could be a max.
A quick second-derivative check finishes it:
S''(10) > 0.
That’s the magic line examiners want. Positive means it’s curving upwards: genuine minimum.
🧠 Let’s Step Away From the Algebra for a Moment
Just imagine the extremes — this is where the “Ohhhh, I get it now” moment happens.
Tiny radius → tall and skinny → massive curved surface
Huge radius → giant circles → surface area explodes again
Too narrow? Bad.
Too wide? Also bad.
There has to be a middle ground — differentiation just finds it cleanly.
❗ Common Errors & Exam Traps
- Forgetting the tank has two circular faces 🧠⚙️
- Rearranging the volume formula incorrectly
- Losing the minus sign when differentiating
- Stating the turning point but not the minimum check
- Examiner-loved line: “Since S''(r) > 0, the stationary point is a minimum.”
🌍 Real-World Link
Companies who build water tanks, grain silos, pressure vessels — they genuinely optimise for material cost. Getting this wrong by even a few percent can cost thousands across a production run.
This is literally the A Level Maths version of industrial design.
🚀 Next Steps
If you want modelling assumptions to become instinct rather than something you remember halfway through the question, the A Level Maths Revision Course that actually works walks through friction, tension, pulleys, rods, moments, projectiles and full mixed-mechanics modelling — slowly, visually, and with real exam-style reasoning rather than panic algebra.
📏 Recap Table
- Volume links radius and height
- Height becomes a function of radius
- Substitute into surface area
- Differentiate once
- Solve for radius
- Use radius to get height
- Check the second derivative
👤Author Bio
S. Mahandru is Head of Maths at Exam.tips and has spent more than 15 years helping students break through the fog of Pure Maths. Known for lively explanations and an honest, straightforward style, he focuses on giving students the confidence to tackle even the hardest exam questions.
🧭 Next step:
Surface-area was just the warm-up. The real puzzle arrives in Advanced Optimisation: Maximum Volume, Minimum Cost, where differentiation meets budgeting and geometry at the same table.
❓ FAQ Section
❓ Would the answer change if the tank had no top?
Totally — one circle disappears from the model, so the ratio changes.
❓ Is eliminating radius instead of height an option?
Technically yes. Practically… it’s a mess. Height is far easier to remove cleanly.
❓ Does the “1:2 ratio” always happen?
Nope — it’s specific to this volume and setup.