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SL 5.7—Optimisation - IB Questionbank | RevisionDojo

Practice SL 5.7—Optimisation with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

A sustainable fashion company produces and sells eco-friendly clothing. The profit, in dollars, from selling $x$ units of a particular clothing item is given by:

$P(x) = 100x - 5x^2$

Determine the number of units $x$ that maximises the profit.

[4]

Calculate the maximum profit.

[3]

Determine whether the function $P(x)$ has a point of inflection, showing all necessary steps.

[8]

Question 2

SLPaper 1

A point $P\left(t, t^{2}-4 t\right)$ lies on the parabola $y=x^{2}-4 x$ . Points $A(2,0)$ and $B(6,0)$ lie on the x-axis. Let $D$ be the sum of distances $P A$ and $P B$ .

Express $D$ in terms of $t$ .

[3]

Let $S=D^{2}$ . Find the derivative $\frac{d S}{d t}$ .

[4]

Find the value of $t$ that minimizes $D$ , and justify that it is a minimum.

[2]

Calculate the minimum sum of distances and the coordinates of $P$ .

[3]

Sketch the parabola and points $A, B, P$ , indicating the minimum distance paths.

[3]

Question 3

SLPaper 2

A composite shape consists of a hemisphere of radius $r \mathrm{~cm}$ attached to the top of a cylinder of radius $r \mathrm{~cm}$ and height $h \mathrm{~cm}$ . The total volume is $500 \pi \mathrm{~cm}^{3}$ , and the cost of material is $\$ 0.05 $per$ \mathrm{cm}^{2} $for the curved surfaces and$ $ 0.03 $per$ \mathrm{cm}^{2}$ for the flat base.

Show that $h=\frac{1500-2 r^{3}}{3 r^{2}}$ .

[3]

Derive an expression for the total cost $C$ in dollars in terms of $r$ .

[4]

Find the value of $r$ that minimizes the cost, and calculate the minimum cost. Justify your answer.

[6]

Calculate the height $h$ and the total surface area at the minimum cost.

[3]

Question 4

SLPaper 2

An open-top tank with a square base is designed to have a total surface area of $108 \text{ m}^2$ . Let the side length of the square base be $x$ metres, and the height of the tank be $h$ metres, as shown in the figure below.

Show that the height of the tank can be expressed as:

$h = \frac{108 - x^2}{4x}$

[5]

Write an expression for the volume $V(x)$ of the tank in terms of $x$ only.

[4]

Use differentiation to find the value of $x$ that maximises the volume of the tank.

[6]

Calculate the height of the tank when the volume is maximised, and hence determine the maximum volume.

[5]

Question 5

SLPaper 2

A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm.

The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area.

Find the slant height of the cone-shaped container.

[2]

Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.

[3]

Find the height, $h$, of this cylinder-shaped container.

[4]

The factory director wants to increase the volume of coconut water sold per container.

State whether or not they should replace the cone-shaped containers with cylinder‑shaped containers. Justify your conclusion.

[4]

Question 6

SLPaper 2

A company produces $x$ units of a product weekly, with revenue modeled by $R(x)= -0.01 x^{2}+50 x$ dollars and cost by $C(x)=0.005 x^{2}+20 x+5000$ dollars. Additionally, the company must store unsold units, with storage cost $S(x)=0.02(x-200)^{2}$ dollars.

Write an expression for the total profit $P(x)$ .

[3]

Find the derivative $\frac{dP}{dx}$ .

[3]

Determine the production level $x$ that maximizes profit, and calculate the maximum profit. Justify your answer.

[6]

If the company can only produce between 100 and 300 units, find the production level that minimizes storage cost and compare its profit to the global maximum found in Part 3.

[4]

Question 7

SLPaper 1

A company's profit per year was found to be changing at a rate of $\frac{dP}{dt} = 3t^2 - 8t$ where $P$ is the company's profit in thousands of dollars and $t$ is the time since the company was founded, measured in years.

Determine whether the profit is increasing or decreasing when $t = 2$ .

[2]

One year after the company was founded, the profit was $4$ thousand dollars. Find an expression for $P(t)$ , when $t \ge 0$ .

[4]

Question 8

SLPaper 2

A cylindrical water tank has a radius $r \text{ cm}$ and height $h \text{ cm}$ , with a total surface area (including top and bottom) of $200 \pi \text{ cm}^2$ .

Express $h$ in terms of $r$ .

[2]

Show that the volume $V \text{ cm}^3$ of the tank is given by $V = \pi r(100 - r^2)$ .

[2]

Find the radius that maximizes the volume, and calculate the maximum volume. Justify your answer.

[4]

Question 9

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$, for $0 \leqslant t \leqslant 12$, where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$.

[2]

Find the value of $q$.

[2]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Question 10

SLPaper 1

The graphs of $y = 6 - x$ and $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

Calculate the area of the shaded region in Diagram 1.

[3]

Write down an integral for the area of the shaded region in Diagram 2.

[1]

Calculate the area of the shaded region in Diagram 2.

[3]

Hence, determine the area enclosed between the line $y = 6 - x$ and the curve $y = 1.5x^{2} - 2.5x + 3$ .

[1]

Paper

Level

Question 1

SLPaper 1

A sustainable fashion company produces and sells eco-friendly clothing. The profit, in dollars, from selling $x$ units of a particular clothing item is given by:

$P(x) = 100x - 5x^2$

Determine the number of units $x$ that maximises the profit.

[4]

Calculate the maximum profit.

[3]

Determine whether the function $P(x)$ has a point of inflection, showing all necessary steps.

[8]

Question 2

SLPaper 1

A point $P\left(t, t^{2}-4 t\right)$ lies on the parabola $y=x^{2}-4 x$ . Points $A(2,0)$ and $B(6,0)$ lie on the x-axis. Let $D$ be the sum of distances $P A$ and $P B$ .

Express $D$ in terms of $t$ .

[3]

Let $S=D^{2}$ . Find the derivative $\frac{d S}{d t}$ .

[4]

Find the value of $t$ that minimizes $D$ , and justify that it is a minimum.

[2]

Calculate the minimum sum of distances and the coordinates of $P$ .

[3]

Sketch the parabola and points $A, B, P$ , indicating the minimum distance paths.

[3]

Question 3

SLPaper 2

Show that $h=\frac{1500-2 r^{3}}{3 r^{2}}$ .

[3]

Derive an expression for the total cost $C$ in dollars in terms of $r$ .

[4]

Find the value of $r$ that minimizes the cost, and calculate the minimum cost. Justify your answer.

[6]

Calculate the height $h$ and the total surface area at the minimum cost.

[3]

Question 4

SLPaper 2

Show that the height of the tank can be expressed as:

$h = \frac{108 - x^2}{4x}$

[5]

Write an expression for the volume $V(x)$ of the tank in terms of $x$ only.

[4]

Use differentiation to find the value of $x$ that maximises the volume of the tank.

[6]

Calculate the height of the tank when the volume is maximised, and hence determine the maximum volume.

[5]

Question 5

SLPaper 2

A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm.

The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area.

Find the slant height of the cone-shaped container.

[2]

Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.

[3]

Find the height, $h$, of this cylinder-shaped container.

[4]

The factory director wants to increase the volume of coconut water sold per container.

State whether or not they should replace the cone-shaped containers with cylinder‑shaped containers. Justify your conclusion.

[4]

Question 6

SLPaper 2

Write an expression for the total profit $P(x)$ .

[3]

Find the derivative $\frac{dP}{dx}$ .

[3]

Determine the production level $x$ that maximizes profit, and calculate the maximum profit. Justify your answer.

[6]

If the company can only produce between 100 and 300 units, find the production level that minimizes storage cost and compare its profit to the global maximum found in Part 3.

[4]

Question 7

SLPaper 1

Determine whether the profit is increasing or decreasing when $t = 2$ .

[2]

One year after the company was founded, the profit was $4$ thousand dollars. Find an expression for $P(t)$ , when $t \ge 0$ .

[4]

Question 8

SLPaper 2

A cylindrical water tank has a radius $r \text{ cm}$ and height $h \text{ cm}$ , with a total surface area (including top and bottom) of $200 \pi \text{ cm}^2$ .

Express $h$ in terms of $r$ .

[2]

Show that the volume $V \text{ cm}^3$ of the tank is given by $V = \pi r(100 - r^2)$ .

[2]

Find the radius that maximizes the volume, and calculate the maximum volume. Justify your answer.

[4]

Question 9

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$, for $0 \leqslant t \leqslant 12$, where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$.

[2]

Find the value of $q$.

[2]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Question 10

SLPaper 1

The graphs of $y = 6 - x$ and $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

Calculate the area of the shaded region in Diagram 1.

[3]

Write down an integral for the area of the shaded region in Diagram 2.

[1]

Calculate the area of the shaded region in Diagram 2.

[3]

Hence, determine the area enclosed between the line $y = 6 - x$ and the curve $y = 1.5x^{2} - 2.5x + 3$ .

[1]

Paper

Level

Question 1

SLPaper 1

A sustainable fashion company produces and sells eco-friendly clothing. The profit, in dollars, from selling $x$ units of a particular clothing item is given by:

$P(x) = 100x - 5x^2$

Determine the number of units $x$ that maximises the profit.

[4]

Calculate the maximum profit.

[3]

Determine whether the function $P(x)$ has a point of inflection, showing all necessary steps.

[8]

Question 2

SLPaper 1

A point $P\left(t, t^{2}-4 t\right)$ lies on the parabola $y=x^{2}-4 x$ . Points $A(2,0)$ and $B(6,0)$ lie on the x-axis. Let $D$ be the sum of distances $P A$ and $P B$ .

Express $D$ in terms of $t$ .

[3]

Let $S=D^{2}$ . Find the derivative $\frac{d S}{d t}$ .

[4]

Find the value of $t$ that minimizes $D$ , and justify that it is a minimum.

[2]

Calculate the minimum sum of distances and the coordinates of $P$ .

[3]

Sketch the parabola and points $A, B, P$ , indicating the minimum distance paths.

[3]

Question 3

SLPaper 2

Show that $h=\frac{1500-2 r^{3}}{3 r^{2}}$ .

[3]

Derive an expression for the total cost $C$ in dollars in terms of $r$ .

[4]

Find the value of $r$ that minimizes the cost, and calculate the minimum cost. Justify your answer.

[6]

Calculate the height $h$ and the total surface area at the minimum cost.

[3]

Question 4

SLPaper 2

Show that the height of the tank can be expressed as:

$h = \frac{108 - x^2}{4x}$

[5]

Write an expression for the volume $V(x)$ of the tank in terms of $x$ only.

[4]

Use differentiation to find the value of $x$ that maximises the volume of the tank.

[6]

Calculate the height of the tank when the volume is maximised, and hence determine the maximum volume.

[5]

Question 5

SLPaper 2

A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm.

The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area.

Find the slant height of the cone-shaped container.

[2]

Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.

[3]

Find the height, $h$, of this cylinder-shaped container.

[4]

The factory director wants to increase the volume of coconut water sold per container.

State whether or not they should replace the cone-shaped containers with cylinder‑shaped containers. Justify your conclusion.

[4]

Question 6

SLPaper 2

Write an expression for the total profit $P(x)$ .

[3]

Find the derivative $\frac{dP}{dx}$ .

[3]

Determine the production level $x$ that maximizes profit, and calculate the maximum profit. Justify your answer.

[6]

If the company can only produce between 100 and 300 units, find the production level that minimizes storage cost and compare its profit to the global maximum found in Part 3.

[4]

Question 7

SLPaper 1

Determine whether the profit is increasing or decreasing when $t = 2$ .

[2]

One year after the company was founded, the profit was $4$ thousand dollars. Find an expression for $P(t)$ , when $t \ge 0$ .

[4]

Question 8

SLPaper 2

A cylindrical water tank has a radius $r \text{ cm}$ and height $h \text{ cm}$ , with a total surface area (including top and bottom) of $200 \pi \text{ cm}^2$ .

Express $h$ in terms of $r$ .

[2]

Show that the volume $V \text{ cm}^3$ of the tank is given by $V = \pi r(100 - r^2)$ .

[2]

Find the radius that maximizes the volume, and calculate the maximum volume. Justify your answer.

[4]

Question 9

SLPaper 2

The depth of water in a port is modelled by the function $d(t) = p\cos qt + 7.5$, for $0 \leqslant t \leqslant 12$, where $t$ is the number of hours after high tide.

At high tide, the depth is 9.7 metres.

At low tide, which is 7 hours later, the depth is 5.3 metres.

Find the value of $p$.

[2]

Find the value of $q$.

[2]

Use the model to find the depth of the water 10 hours after high tide.

[2]

Question 10

SLPaper 1

The graphs of $y = 6 - x$ and $y = 1.5x^{2} - 2.5x + 3$ intersect at $(-1, 7)$ and $(2, 4)$ , as shown in the following diagrams.

In Diagram 1, the region enclosed by the line $y = 6 - x$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 1

In Diagram 2, the region enclosed by the curve $y = 1.5x^{2} - 2.5x + 3$ , the vertical lines $x = -1$ and $x = 2$ , and the $x$ -axis has been shaded.

Diagram 2

Calculate the area of the shaded region in Diagram 1.

[3]

Write down an integral for the area of the shaded region in Diagram 2.

[1]

Calculate the area of the shaded region in Diagram 2.

[3]

Hence, determine the area enclosed between the line $y = 6 - x$ and the curve $y = 1.5x^{2} - 2.5x + 3$ .

[1]

SL 5.7—Optimisation Questionbank