IB Questionbank

Updates to Questionbank

User interface language: English | Español

Date	November 2016	Marks available	2	Reference code	16N.2.SL.TZ0.T_6
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Show that	Question number	T_6	Adapted from	N/A

Question

A water container is made in the shape of a cylinder with internal height $h$ cm and internal base radius $r$ cm.

N16/5/MATSD/SP2/ENG/TZ0/06

The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.

The volume of the water container is $0.5{\text{ }}{{\text{m}}^3}$ .

The water container is designed so that the area to be coated is minimized.

One can of water-resistant material coats a surface area of $2000{\text{ c}}{{\text{m}}^2}$ .

Write down a formula for $A$ , the surface area to be coated.

[2]

a.

Express this volume in ${\text{c}}{{\text{m}}^3}$ .

[1]

b.

Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.

[1]

c.

Show that $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ .

[2]

d.

Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .

[3]

e.

Using your answer to part (e), find the value of $r$ which minimizes $A$ .

[3]

f.

Find the value of this minimum area.

[2]

g.

Find the least number of cans of water-resistant material that will coat the area in part (g).

[3]

h.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$(A = ){\text{ }}\pi {r^2} + 2\pi rh$ (A1)(A1)

Note: Award (A1) for either $\pi {r^2}$ OR $2\pi rh$ seen. Award (A1) for two correct terms added together.

[2 marks]

a.

$500\,000$ (A1)

Notes: Units not required.

[1 mark]

b.

$500\,000 = \pi {r^2}h$ (A1)(ft)

Notes: Award (A1)(ft) for $\pi {r^2}h$ equating to their part (b).

Do not accept unless $V = \pi {r^2}h$ is explicitly defined as their part (b).

[1 mark]

c.

$A = \pi {r^2} + 2\pi r\left( {\frac{{500\,000}}{{\pi {r^2}}}} \right)$ (A1)(ft)(M1)

Note: Award (A1)(ft) for their ${\frac{{500\,000}}{{\pi {r^2}}}}$ seen.

Award (M1) for correctly substituting only ${\frac{{500\,000}}{{\pi {r^2}}}}$ into a correct part (a).

Award (A1)(ft)(M1) for rearranging part (c) to $\pi rh = \frac{{500\,000}}{r}$ and substituting for $\pi rh$ in expression for $A$ .

$A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ (AG)

Notes: The conclusion, $A = \pi {r^2} + \frac{{1\,000\,000}}{r}$ , must be consistent with their working seen for the (A1) to be awarded.

Accept ${10^6}$ as equivalent to ${1\,000\,000}$ .

[2 marks]

d.

$2\pi r - \frac{{{\text{1}}\,{\text{000}}\,{\text{000}}}}{{{r^2}}}$ (A1)(A1)(A1)

Note: Award (A1) for $2\pi r$ , (A1) for $\frac{1}{{{r^2}}}$ or ${r^{ - 2}}$ , (A1) for $- {\text{1}}\,{\text{000}}\,{\text{000}}$ .

[3 marks]

e.

$2\pi r - \frac{{1\,000\,000}}{{{r^2}}} = 0$ (M1)

Note: Award (M1) for equating their part (e) to zero.

${r^3} = \frac{{1\,000\,000}}{{2\pi }}$ OR $r = \sqrt[3]{{\frac{{1\,000\,000}}{{2\pi }}}}$ (M1)

Note: Award (M1) for isolating $r$ .

OR

sketch of derivative function (M1)

with its zero indicated (M1)

$(r = ){\text{ }}54.2{\text{ }}({\text{cm}}){\text{ }}(54.1926 \ldots )$ (A1)(ft)(G2)

[3 marks]

f.

$\pi {(54.1926 \ldots )^2} + \frac{{1\,000\,000}}{{(54.1926 \ldots )}}$ (M1)

Note: Award (M1) for correct substitution of their part (f) into the given equation.

$= 27\,700{\text{ }}({\text{c}}{{\text{m}}^2}){\text{ }}(27\,679.0 \ldots )$ (A1)(ft)(G2)

[2 marks]

g.

$\frac{{27\,679.0 \ldots }}{{2000}}$ (M1)

Note: Award (M1) for dividing their part (g) by 2000.

$= 13.8395 \ldots$ (A1)(ft)

Notes: Follow through from part (g).

14 (cans) (A1)(ft)(G3)

Notes: Final (A1) awarded for rounding up their $13.8395 \ldots$ to the next integer.

[3 marks]

h.

Examiners report

[N/A]

a.

[N/A]

b.

[N/A]

c.

[N/A]

d.

[N/A]

e.

[N/A]

f.

[N/A]

g.

[N/A]

h.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.1—3d space, volume, angles, distance, midpoints

Show 73 related questions

18N.1.SL.TZ0.T_1b:
Write down your answer to part (a) correct to the nearest integer.
17M.2.SL.TZ1.T_4f:
In the context of this model, state what the value of 19 represents.
22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ lies on both $\prod_{1}$ and $\prod_{2}$ .
18N.1.SL.TZ0.T_9a.i:
Write down the value of x.
22M.2.SL.TZ1.7c:
Find the height of the pyramid, $VX$ .
22M.2.SL.TZ1.7d:
A second ornament is in the shape of a cuboid with a rectangular base of length $2 x cm$ , width $x cm$ and height $y cm$ . The cuboid has the same volume as the pyramid.

The cuboid has a minimum surface area of $S {cm}^{2}$ . Find the value of $S$ .
17M.2.SL.TZ1.T_4c:
Find the value of $a$ .
18M.2.SL.TZ1.T_1d:
Calculate the length of AE.
SPM.2.SL.TZ0.1b:
The sphere is to be melted down and remoulded into the shape of a cone with a height of 14.8 cm.

Find the radius of the base of the cone, correct to 2 significant figures.
16N.1.SL.TZ0.T_7b:
Calculate the radius of the balloon following this increase.
22M.1.AHL.TZ1.11b.ii:
Find a vector equation of $L$ , the line of intersection of $\prod_{1}$ and $\prod_{2}$ .
22M.1.AHL.TZ1.11c:
Find the distance between $L$ and $\prod_{3}$ .
17N.2.SL.TZ0.T_6c:
Find the cost of $100{\text{ c}}{{\text{m}}^3}$ of chocolate mousse.
21M.1.SL.TZ1.2a:
Write down the radius of the planet.
21M.1.SL.TZ1.2b:
The volume of the planet can be expressed in the form $π (a \times 10^{k}) {km}^{3}$ where $1 \leq a < 10$ and $k \in ℤ$ .

Find the value of $a$ and the value of $k$ .
21M.1.SL.TZ2.8d:
Hence, find the volume of the cone.
22M.1.AHL.TZ1.11a:
Show that the three planes do not intersect.
22M.2.SL.TZ1.7a:
Find $AV$ .
18M.2.SL.TZ1.T_1c:
Calculate the length of MN.
19M.2.SL.TZ1.T_2c:
Find the maximum number of complete panels that can be fitted to the whole roof.
17N.2.SL.TZ0.T_6f:
Find the value of $x$ .
19M.1.SL.TZ1.T_15a:
Write down an equation for the area, $A$ , of the curved surface in terms of $r$ .
19M.2.SL.TZ2.T_3b:
Find the slant height of the cone-shaped container.
18N.1.SL.TZ0.T_1a:
Calculate the volume of this hemisphere in cm³.

Give your answer correct to one decimal place.
17M.1.SL.TZ1.T_9b:
Find the slant height, $l$ , of the cone.
20N.1.SL.TZ0.T_1b:
Calculate James’s estimate of its volume, in ${km}^{3}$ .
20N.1.SL.TZ0.T_1a:
Write down the value of the iron in the form $a \times 10^{k}$ where $1 \leq a < 10, k \in ℤ$ .
17M.2.SL.TZ1.T_4a:
Calculate the volume of this pan.
16N.2.SL.TZ0.T_6b:
Express this volume in ${\text{c}}{{\text{m}}^3}$ .
16N.2.SL.TZ0.T_6c:
Write down, in terms of $r$ and $h$ , an equation for the volume of this water container.
16N.2.SL.TZ0.T_6e:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
16N.2.SL.TZ0.T_6f:
Using your answer to part (e), find the value of $r$ which minimizes $A$ .
SPM.2.SL.TZ0.1a:
Find the volume of the sphere expressing your answer in the form $a \times {10^k}$ , $1 \leqslant a < 10$ and $k \in \mathbb{Z}$ .
18M.1.SL.TZ1.T_14a:
Calculate the radius of the base of the cone which has been removed.
20N.1.SL.TZ0.T_1c:
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.
18M.1.SL.TZ2.T_15b:
The total external surface area of the pencil case rounded to 3 significant figures is 570 cm².

Using your graphic display calculator, calculate the value of r.
21M.1.SL.TZ2.8c:
Find the value of $l$ .
17N.2.SL.TZ0.T_6b:
Calculate the radius, $r$ , of a hemisphere shaped glass.
18N.1.SL.TZ0.T_1c:
Write down your answer to part (b) in the form a × 10^k , where 1 ≤ a < 10 and k ∈ $\mathbb{Z}$ .
16N.1.SL.TZ0.T_7a:
Calculate the volume of the balloon.
19M.1.SL.TZ1.T_1c:
Find the volume, in cm³, of this calculator case.
18N.1.SL.TZ0.T_9b:
1 cm³ of glass has a mass of 2.56 grams.

Calculate the mass, in grams, of the paperweight.
17M.1.SL.TZ2.T_12b:
Find the value of $h$ .
17M.1.SL.TZ2.T_12a:
Find the volume of water in the container.
19M.1.SL.TZ1.T_15c:
Find the value of $r$ when the area of the curved surface is maximized.
18M.1.SL.TZ2.T_15a:
Find an expression for the slant height of the cone in terms of r.
17N.2.SL.TZ0.T_6a:
Show that the volume of a cone shaped glass is $160{\text{ c}}{{\text{m}}^3}$ , correct to 3 significant figures.
18M.1.SL.TZ1.T_14b:
Calculate the curved surface area of the cone which has been removed.
17M.2.SL.TZ1.T_4d:
Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.
17M.1.SL.TZ1.T_9a:
Find the radius, $r$ , of the circular base of the cone.
17M.1.SL.TZ1.T_9c:
Find the curved surface area of the cone.
18M.1.SL.TZ1.T_14c:
Calculate the curved surface area of the remaining solid.
18M.2.SL.TZ1.T_1e:
Show that Farmer Brown is incorrect.
17N.2.SL.TZ0.T_6d:
Show that there is $20{\text{ c}}{{\text{m}}^3}$ of orange paste in each special dessert.
16N.2.SL.TZ0.T_6g:
Find the value of this minimum area.
19M.1.SL.TZ1.T_15b:
Find $\frac{{{\text{d}}A}}{{{\text{d}}r}}$ .
18N.1.SL.TZ0.T_9a.ii:
Calculate the volume of the paperweight.
19M.2.SL.TZ1.T_2a:
Find the length ${\text{AB}}$ , giving your answer to four significant figures.
19M.2.SL.TZ1.T_2b:
Find the total area of the two rectangles that make up the roof.
19M.2.SL.TZ1.T_2d:
Find the percentage error in her estimate.
19M.2.SL.TZ1.T_2e:
Olivia investigates arranging the panels, such that the longer edge of each panel is parallel to ${\text{AB}}$ or ${\text{AC}}$ .

State whether this new arrangement will allow Olivia to fit more solar panels to the roof. Justify your answer.
17M.2.SL.TZ1.T_4b:
Find the radius of the sphere in cm, correct to one decimal place.
17M.2.SL.TZ1.T_4e:
Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.
16N.2.SL.TZ0.T_6a:
Write down a formula for $A$ , the surface area to be coated.
16N.2.SL.TZ0.T_6h:
Find the least number of cans of water-resistant material that will coat the area in part (g).
19M.2.SL.TZ2.T_3c:
Show that the total surface area of the cone-shaped container is 314 cm², correct to three significant figures.
19M.2.SL.TZ2.T_3d:
Find the height, $h$ , of this cylinder-shaped container.
19M.2.SL.TZ2.T_3a:
Find the volume of one cone-shaped container.
19M.2.SL.TZ2.T_3e:
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.
18M.2.SL.TZ1.T_1a:
Calculate the area of triangle EAD.
18M.2.SL.TZ1.T_1b:
Calculate the total volume of the barn.
18M.2.SL.TZ1.T_1f:
Calculate the total length of metal required for one support.
17N.2.SL.TZ0.T_6e:
Find the total cost of the ingredients of one special dessert.

Hide related questions

Topic 5 —Calculus » SL 5.1—Introduction of differential calculus

Topic 5 —Calculus » SL 5.3—Differentiating polynomials, n E Z

Topic 4—Statistics and probability » SL 4.5—Probability concepts, expected numbers

Topic 5 —Calculus » SL 5.5—Integration introduction, areas between curve and x axis

Topic 4—Statistics and probability » SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams

Topic 5 —Calculus » SL 5.8—Testing for max and min, optimisation. Points of inflexion

Topic 3— Geometry and trigonometry

Topic 4—Statistics and probability

Topic 5 —Calculus