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SL 3.1—3d space, volume, angles, midpoints - IB Questionbank | RevisionDojo

Practice SL 3.1—3d space, volume, angles, midpoints 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

An architect is designing a right pyramid with a square base for a new monument. The base has a side length of 10 cm, and the height from the base to the apex is 12 cm.

Calculate the area of the square base.

[1]

Using your answer from part (a), determine the volume of the pyramid.

[2]

Calculate the slant height of the pyramid, which is the distance from the center of the base to the apex along the pyramid's side.

[2]

Find the total surface area of the pyramid, including the base and all triangular faces.

[1]

Question 2

SLPaper 1

A hot air balloon is directly above a point on the ground. Two observers, A and B, are standing 100 meters apart from each other. The angle of elevation from observer A to the balloon is $40°$ , and the angle of elevation from observer B is $60°$ . The hot air ballon is directly in between person A and B such that the hot air balloon and the two people all lie on a straight line.

Draw a labeled diagram representing the situation, showing the positions of the balloon, observers, and angles of elevation.

[2]

Calculate the height of the balloon above the ground based on the observations from points A and B.

[3]

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 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 4

SLPaper 1

Iron in the asteroid 16 Psyche is said to be valued at $8973$ quadrillion euros $(EUR)$ ,where one quadrillion $= 10^{15}$ .

James believes the asteroid is approximately spherical with radius $113 km$ . He uses thisinformation to estimate its volume.

Write down the value of the iron in the form $a \times 10^{k}$ where $1 \leq a < 10, k \in ℤ$ .

[2]

Calculate James’s estimate of its volume, in ${km}^{3}$ .

[2]

The actual volume of the asteroid is found to be $6.074 \times 10^{6} {km}^{3}$ .

Find the percentage error in James’s estimate of the volume.

[2]

Question 5

SLPaper 1

Joey is making a party hat in the form of a cone. The hat is made from a sector, AOB, of a circular piece of paper with a radius of 18 cm and AOB = θ as shown in the diagram. To make the hat, sides [OA] and [OB] are joined together. The hat has a base radius of 6.5 cm.

Write down the perimeter of the base of the hat in terms of π.

[1]

Find the value of θ.

[2]

Find the surface area of the outside of the hat.

[2]

Question 6

HLPaper 1

A civil engineer is studying the intersection of two pathways in a park, represented by the direction vectors ${\mathbf{a} = \langle 3, 4, 0 \rangle}$ and ${\mathbf{b} = \langle 1, -1, 2 \rangle}$ .

Calculate the angle between these pathways.

[3]

Determine the angle between the plane defined by the equation ${2x - y + z = 5}$ and the line described by the parametric equations ${x = 2t, y = 1 - t, z = 3 + t}$ . Explain what this angle indicates about the position of the line relative to the plane.

[4]

Question 7

SLPaper 1

A balloon in the shape of a sphere is filled with helium until the radius is 6 cm.

The volume of the balloon is increased by 40%.

Calculate the volume of the balloon.

[2]

Calculate the radius of the balloon following this increase.

[4]

Question 8

SLPaper 1

A cylindrical water tank with a radius of 5 cm and a height of 12 cm is topped with a conical funnel of the same radius and a height of 9 cm to aid in filling the tank.

Calculate the total volume of the combined shape, which represents the maximum capacity of the tank with the funnel.

[3]

Determine the total surface area of the combined shape, including the cylinder and the cone, to understand the material required for manufacturing.

[3]

Question 9

HLPaper 1

A submarine is located in a sea at coordinates $(0.8, 1.3, - 0.3)$ relative to a ship positionedat the origin $O$ . The $x$ direction is due east, the $y$ direction is due north and the $z$ direction isvertically upwards.

All distances are measured in kilometres.

The submarine travels with direction vector $(\begin{matrix} - 2 \\ - 3 \\ 1 \end{matrix})$ .

The submarine reaches the surface of the sea at the point $P$ .

Assuming the submarine travels in a straight line, write down an equation for the linealong which it travels.

[2]

Find the coordinates of $P$ .

[3]

Find $OP$ .

[2]

Question 10

HLPaper 1

Find the coordinates of the point of intersection of the planes defined by the equations $x + y + z = 3,{\text{ }}x - y + z = 5$ and $x + y + 2z = 6$ .

SL 3.1—3d space, volume, angles, midpoints Questionbank