IB Mathematics Applications & Interpretation Topic SL 5.7—optimisation Questionbank (SL/HL)

IB Mathematics Applications & Interpretation Topic SL 5.7—optimisation Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Applications & Interpretation (AI) Topic SL 5.7—optimisation with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.7—optimisation and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 2

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

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

[3]

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

[4]

Use technology to find the value of $r$ that minimizes the surface area, and calculate the minimum surface area. Justify that this gives a minimum.

[6]

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

[3]

Sketch the frustum and label the dimensions corresponding to the minimum surface area. radius

r

radius

2 r

[3]

Question 2

SLPaper 2

An open-top tank with a square base is designed to have a total surface area of $108 m^{2}$ . 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:

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

[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 3

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 the cylinder-shaped container with the same radius and total surface area as the cone-shaped container.

[4]

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

State whether they should replace the cone-shaped containers with cylinder-shaped containers. Justify your conclusion.

[4]

Question 4

SLPaper 2

A chocolatier is designing a closed cylindrical gift tin that must hold a volume of $500 cm^{3}$ of chocolates. The metal for the lid and base costs 2 cents per square centimetre, while the metal for the curved side costs 1 cent per square centimetre. Let $x$ and $y$ be the radius and height, in cm, and let $K$ be the total cost in cents. The company wants to minimize the manufacturing cost of the tin.

Find an expression for the total cost $K$ of the tin in terms of the radius $x$ and height $y$ .

[3]

Given the volume is $500 cm^{3}$ , find $y$ in terms of $x$ .

[2]

Find the radius $x$ that minimizes the total cost.

[5]

Question 5

SLPaper 2

A school is designing an open-topped planter box with a square base that must hold $441 dm^{3}$ of soil. The material for the base costs 18 cents per square decimetre and the material for the four sides costs 7 cents per square decimetre. Let the side length of the base be $x$ dm, the height be $h$ dm, and the total cost be $C$ cents.

Find an expression for the total cost $C$ of the planter box in terms of $x$ and $h$ .

[3]

Given the volume is $441 dm^{3}$ , find $h$ in terms of $x$ .

[2]

Find the value of $x$ that minimizes the total cost.

[5]

IB Mathematics Applications & Interpretation SL 5.7—optimisation Questionbank

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IB Mathematics Applications & Interpretation SL 5.7—optimisation Questionbank

IB Mathematics Applications & Interpretation (AI) Topic SL 5.7—optimisation Questionbank

Unlock all 31 questions with a free account