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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 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 if $x \neq 200$ .

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

[3]

Find the derivative $\frac{d P}{d x}$ .

[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 maximum.

[4]

Question 3

SLPaper 2

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

Express $h$ in terms of $r$ .

[2]

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

[2]

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

[4]

Question 4

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 5

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 6

SLPaper 2

A rectangular field is enclosed by 120 m of fencing. One side of the field is along a river and requires no fencing. Let the side along the river have length $x \mathrm{~m}$ .

Express the area $A \mathrm{~m}^{2}$ of the field in terms of $x$ .

[2]

Find the dimensions of the field that maximize the area, and calculate the maximum area.

[4]

Sketch the field and its fencing, indicating the dimensions for maximum area.

[2]

Question 7

SLPaper 2

A conical frustum is designed as a water reservoir with a smaller circular top of radius $r \mathrm{~m}$ , a larger circular base of radius $2 r \mathrm{~m}$ , and height $h \mathrm{~m}$ . The volume of the frustum is fixed at $100 \pi \mathrm{~m}^{3}$ .

Show that the height $h$ can be expressed as $h=\frac{300}{7 r^{2}}$ .

[3]

Derive an expression for the total surface area $S \mathrm{~m}^{2}$ (including top and bottom) in terms of $r$ .

[4]

Find the value of $r$ that minimizes the surface area, and calculate the minimum surface area. Justify your answer using calculus.

[6]

Determine the slant height of the frustum at the minimum surface area.

[3]

Sketch the frustum, labeling the dimensions for minimum surface area. radius $r$ radius $2 r$

[3]

Question 8

SLPaper 2

Consider the function $f(x)=12-x^{2}$ , for $x \in \mathbb{R}$ . A rectangle PQRS has vertex P at the origin, Q on the positive x -axis, R on the graph of $f$ , and S on the y -axis.

If Q has coordinates ( $x, 0$ ), express the area $A$ of rectangle PQRS in terms of $x$ .

[2]

Find the value of $x$ that maximizes the area, and state the maximum area.

[3]

Sketch the graph of $f(x)$ and shade the rectangle PQRS for the maximum area.

[2]

Question 9

SLPaper 1

A rectangular garden has a perimeter of $28 \mathrm{~m} . A$ )

Complete the table below by finding the missing values for the dimensions and area of the garden.

Length (m)	Width (m)	Area $\left(\mathrm{m}^{2}\right)$
2	12	24
5	$a$	$b$
$c$	9	45

[2]

If the length of the garden is $l \mathrm{~m}$ , express the area $A$ in terms of $l$ only.

[1]

Find the dimensions of the garden that maximize the area.

[3]

Question 10

SLPaper 1

A point $P\left(a, a^{2}\right)$ lies on the curve $y=x^{2}$ . Point $A(2,0)$ lies on the x-axis.

Express the distance $D$ between $P$ and $A$ in terms of $a$ .

[2]

Let $S=D^{2}$ . Find the value of $a$ that minimizes $S$ , and justify that it is a minimum.

[4]

Calculate the minimum distance $D$ and the coordinates of $P$ .

[2]

SL 5.7—Optimisation Questionbank