Free Optimisation questions for IB Mathematics: Applications & Interpretation (SL & HL), part of Calculus. Each question has the answer and the marking points an examiner would look for.
Generate a full Optimisation worksheet →All Calculus questionsAn open box has a square base of side x cm and height h cm. Its volume is 256 cm3.
Show that the surface area is S = x2 + 1024x.
Find the value of x that minimizes S and the minimum surface area.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | shown h = 256x2 A1 S = x2 + 4xh AG A1 | 2 |
| (b) | x = 8.00, S = 192 S′ = 2x − 1024x2 A1 S′ = 0 (M1) x = 8.00 A1 S = 192 cm2 A1 | 4 |
A closed cylindrical can holds 500 cm3.
Show that the surface area is A = 2πr2 + 2Vr.
Find the radius that minimizes the surface area, and the minimum area.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | shown h = V/(πr2) M1 substitution A1 | 2 |
| (b) | r = 4.30 cm; 349 cm2 A′(r) = 0 or GDC (M1) r = 4.30 A1 349 A1 | 3 |
The weekly profit, in dollars, from selling x hundred items is modelled by P(x) = −3x2 + 120x − 200.
Find P′(x).
Find the number of items that maximizes the profit, and the maximum profit.
Find the least number of items that must be sold to make a profit.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | −6x + 120 −6x + 120 A1 | 1 |
| (b) | 2000 items; $1000 P′(x) = 0 (M1) x = 20.0 (hundred) A1 $1000 A1 | 3 |
| (c) | 175 items P(x) = 0 (M1) x = 1.74, so 175 items A1 | 2 |
An open box has a square base of side x cm and height h cm. Its volume is 500 cm3.
Show that the surface area is S = x2 + 2000x.
Find the value of x that minimizes S and the minimum surface area.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | shown h = 500x2 A1 S = x2 + 4xh AG A1 | 2 |
| (b) | x = 10.0, S = 300 S′ = 2x − 2000x2 A1 S′ = 0 (M1) x = 10.0 A1 S = 300 cm2 A1 | 4 |
A closed cylindrical can holds 1000 cm3.
Show that the surface area is A = 2πr2 + 2Vr.
Find the radius that minimizes the surface area, and the minimum area.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | shown h = V/(πr2) M1 substitution A1 | 2 |
| (b) | r = 5.42 cm; 554 cm2 A′(r) = 0 or GDC (M1) r = 5.42 A1 554 A1 | 3 |
The weekly profit, in dollars, from selling x hundred items is modelled by P(x) = −3x2 + 96x − 400.
Find P′(x).
Find the number of items that maximizes the profit, and the maximum profit.
Find the least number of items that must be sold to make a profit.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | −6x + 96 −6x + 96 A1 | 1 |
| (b) | 1600 items; $368 P′(x) = 0 (M1) x = 16.0 (hundred) A1 $368 A1 | 3 |
| (c) | 493 items P(x) = 0 (M1) x = 4.92, so 493 items A1 | 2 |
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Meet Riyaz →Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in