IB Mathematics Applications & Interpretation Topic SL…

A company stores cold brew concentrate in conical canisters that have a radius of 7 cm and a vertical height of 24 cm. Designers are evaluating whether these conical canisters can be replaced with cylindrical ones that have the same base radius and the same total surface area.

(a) the slant height of the conical canister.

(b) the volume of the conical canister.

(c) Verify that the total surface area of the conical canister is $704\text{ cm}^2$ , rounded to three significant figures.

(e) A spherical light of radius $3.5$ m is centred at $M$ . whether it fits entirely inside the atrium. Provide a justification.

(e) The company wants to maximize the liquid volume held in each container. whether the company should transition to cylindrical canisters. Provide a justification for your answer.

A rectangular atrium has opposite corners $O(0,0,0)$ and $G(6,8,24)$ metres. A cable runs along the diagonal $OG$ , and a spherical light is centred at the midpoint $M$ of $OG$ .

(b) the volume of the atrium.

A decorative paperweight is formed by joining the circular base of a right cone to the flat face of a hemisphere. Both radii are $r$ cm and the perpendicular height of the cone is $2r$ cm. The volume of the solid is $288\pi$ cm $^3$ .

Give non-exact numerical answers correct to three significant figures.

(b) Hence the exposed surface area of the solid, explaining which circular faces are excluded.

(c) A sphere has the same volume as the solid. which object has the smaller exposed surface area and the percentage difference relative to the sphere.

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.

(a)

Find the slant height of the cone-shaped container.

(b)

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

(c)

Find the height, $h$ , of the cylinder-shaped container with the same radius and total surface area as the cone-shaped container.

(d)

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.

A glass ornament is a tetrahedron with vertices $A(0,0,0)$ , $B(8,0,0)$ , $C(0,6,0)$ and $D(0,0,9)$ , where coordinates are in cm. $M$ is the midpoint of $BC$ .

(a) the volume of the tetrahedron.

(d) the height, $h$ , of the cylindrical canister described.

(a) the length of $OG$ .

(d) the coordinates of $M$ and the distance from $M$ to the floor.

(a) $r$ and the slant height of the cone.

(b) the angle between $DM$ and plane $ABC$ .

IB Mathematics Applications & Interpretation SL 3.1—3d Space, Volume, Angles, Midpoints Questionbank

IB Mathematics Applications & Interpretation SL 3.1—3d Space, Volume, Angles, Midpoints Questionbank

IB Mathematics Applications & Interpretation SL 3.1—3d Space, Volume, Angles, Midpoints Questionbank

IB Mathematics Applications & Interpretation SL 3.1—3d Space, Volume, Angles, Midpoints Questionbank

IB Mathematics Applications & Interpretation (AI) Topic SL 3.1—3d Space, Volume, Angles, Midpoints Questionbank