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Date	November 2018	Marks available	3	Reference code	18N.2.SL.TZ0.S_10
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	S_10	Adapted from	N/A

Question

All lengths in this question are in metres.

Consider the function $f\left( x \right) = \sqrt {\frac{{4 - {x^2}}}{8}}$ , for −2 ≤ $x$ ≤ 2. In the following diagram, the shaded region is enclosed by the graph of $f$ and the $x$ -axis.

A container can be modelled by rotating this region by 360˚ about the $x$ -axis.

Water can flow in and out of the container.

The volume of water in the container is given by the function $g\left( t \right)$ , for 0 ≤ $t$ ≤ 4 , where $t$ is measured in hours and $g\left( t \right)$ is measured in m³. The rate of change of the volume of water in the container is given by $g'\left( t \right) = 0.9 - 2.5\,{\text{cos}}\left( {0.4{t^2}} \right)$ .

The volume of water in the container is increasing only when $p$ < $t$ < $q$ .

Find the volume of the container.

[3]

a.

Find the value of $p$ and of $q$ .

[3]

b.i.

During the interval $p$ < $t$ < $q$ , he volume of water in the container increases by $k$ m³. Find the value of $k$ .

[3]

b.ii.

When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.

[5]

c.

Markscheme

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

attempt to substitute correct limits or the function into formula involving ${f^2}$ (M1)

eg $\pi \int_{ - 2}^2 {{y^2}\,{\text{d}}y}$ , $\pi {\int {\left( {\sqrt {\frac{{4 - {x^2}}}{8}} } \right)} ^2}{\text{d}}x$

4.18879

volume = 4.19, $\frac{4}{3}\pi$ (exact) (m³) A2 N3

Note: If candidates have their GDC incorrectly set in degrees, award M marks where appropriate, but no A marks may be awarded. Answers from degrees are p = 13.1243 and q = 26.9768 in (b)(i) and 12.3130 or 28.3505 in (b)(ii).

[3 marks]

a.

recognizing the volume increases when ${g'}$ is positive (M1)

eg $g'\left( t \right)$ > 0, sketch of graph of ${g'}$ indicating correct interval

1.73387, 3.56393

$p$ = 1.73, $p$ = 3.56 A1A1 N3

[3 marks]

b.i.

valid approach to find change in volume (M1)

eg $g\left( q \right) - g\left( p \right)$ , $\int_p^q {g'\left( t \right){\text{d}}t}$

3.74541

total amount = 3.75 (m³) A2 N3

[3 marks]

b.ii.

Note: There may be slight differences in the final answer, depending on which values candidates carry through from previous parts. Accept answers that are consistent with correct working.

recognizing when the volume of water is a maximum (M1)

eg maximum when $t = q$ , $\int_0^q {g'\left( t \right){\text{d}}t}$

valid approach to find maximum volume of water (M1)

eg $2.3 + \int_0^q {g'\left( t \right){\text{d}}t}$ , $2.3 + \int_0^p {g'\left( t \right){\text{d}}t} + 3.74541$ , 3.85745

correct expression for the difference between volume of container and maximum value (A1)

eg $4.18879 - \left( {2.3 + \int_0^q {g'\left( t \right){\text{d}}t} } \right)$ , 4.19 − 3.85745

0.331334

0.331 (m³) A2 N3

[5 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.

Syllabus sections

Topic 5 —Calculus » AHL 5.17—Areas under curve onto y-axis, volume of revolution (about x and y axes)

Show 36 related questions

22M.2.AHL.TZ1.10c.ii:
Hence, determine the maximum volume of the container.
22M.2.AHL.TZ2.6:
The following diagram shows the curve $\frac{x^{2}}{36} + \frac{{(y - 4)}^{2}}{16} = 1$ , where $h \leq y \leq 4$ .

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The bowl is assumed to have negligible thickness.

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19N.2.SL.TZ0.S_8d:
Let $R$ be the region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = p$ and $x = b$ . The region $R$ is rotated 360º about the $x$ -axis. Find the volume of the solid formed.
19N.2.SL.TZ0.S_8c.ii:
Find the rate of change of $f$ at ${\text{B}}$ .
18N.2.SL.TZ0.S_10c:
When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.
19N.2.SL.TZ0.S_8b.ii:
Find the equation of the tangent to the graph of $f$ at ${\text{A}}$ .
19N.2.SL.TZ0.S_8a:
Find the value of $p$ .
18N.2.SL.TZ0.S_10a:
Find the volume of the container.
17N.2.SL.TZ0.S_5b:
The following diagram shows part of the graph of $f$ .

The region enclosed by the graph of $f$ , the $x$ -axis and the lines $x = - p$ and $x = p$ is rotated 360° about the $x$ -axis. Find the volume of the solid formed.
19N.2.SL.TZ0.S_8b.i:
Write down the coordinates of ${\text{A}}$ .
17N.2.SL.TZ0.S_5a:
Find the value of $p$ .
17M.2.AHL.TZ2.H_2b:
Find the volume of the solid formed when the region bounded by the curve, the $x$ -axis for $x \geqslant 0$ and the $y$ -axis for $y \geqslant 0$ is rotated through $2\pi$ about the $x$ -axis.
22M.2.AHL.TZ1.10c.i:
Show that the volume, $V m^{3}$ , of water in the container when it is filled to a height of $h$ metres is given by $V = π (\frac{1}{3} h^{3} + h)$ .
20N.1.AHL.TZ0.H_12b:
State the equation of the horizontal asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12d:
Sketch the graph of $y = f (x)$ , stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.
20N.2.AHL.TZ0.H_3b:
Hence find the area of the shaded region.
20N.1.SL.TZ0.S_3a:
Find the value of $a$ .
21M.2.AHL.TZ2.11d.ii:
State the cross-sectional radius of the bowl at this point.
21M.2.AHL.TZ2.11d.i:
By sketching the graph of a suitable derivative of $f$ , find where the cross-sectional radius of the bowl is decreasing most rapidly.
16N.2.SL.TZ0.S_6a:
Use the model to find the volume of the barrel.
20N.2.AHL.TZ0.H_3a:
Determine the values of $p$ , $q$ and $r$ .
19N.2.SL.TZ0.S_8c.i:
Find the coordinates of ${\text{B}}$ .
19N.2.AHL.TZ0.H_11c:
The region $R$ is now rotated about the $y$ -axis, through $2\pi$ radians, to form a solid.

By writing ${{\text{si}}{{\text{n}}^3}\,y}$ as $\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y$ , show that the volume of the solid formed is $\frac{{2\pi }}{3}$ .
20N.1.SL.TZ0.S_3b:
Find the volume of the solid formed when the shaded region is revolved $360 °$ about the $x$ -axis.
16N.2.SL.TZ0.S_6b:
The empty barrel is being filled with water. The volume $V{\text{ }}{{\text{m}}^3}$ of water in the barrel after $t$ minutes is given by $V = 0.8(1 - {{\text{e}}^{ - 0.1t}})$ . How long will it take for the barrel to be half-full?
21M.2.AHL.TZ2.11c.i:
Find $OA$ .
20N.1.AHL.TZ0.H_12a:
State the equation of the vertical asymptote on the graph of $y = f (x)$ .
20N.1.AHL.TZ0.H_12c:
Use an algebraic method to determine whether $f$ is a self-inverse function.
19N.1.AHL.TZ0.H_10c:
Show that $\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}$ .
20N.1.AHL.TZ0.H_12e:
The region bounded by the $x$ -axis, the curve $y = f (x)$ , and the lines $x = 5$ and $x = 7$ is rotated through $2 π$ about the $x$ -axis. Find the volume of the solid generated, giving your answer in the form $π (a + b \ln 2)$ , where $a, b \in ℤ$ .
19N.2.AHL.TZ0.H_11b:
Find the area of $R$ .
18N.2.SL.TZ0.S_10b.ii:
During the interval $p$ < $t$ < $q$ , he volume of water in the container increases by $k$ m³. Find the value of $k$ .
19N.1.AHL.TZ0.H_10d:
The area enclosed by the graph of $y = f\left( x \right)$ and the line $y = 4$ can be expressed as ${\text{ln}}\,v$ . Find the value of $v$ .
21M.2.AHL.TZ2.11a:
Show that the volume of the solid formed is $\frac{15 k^{2} π}{34}$ cubic units.
21M.2.AHL.TZ2.11c.ii:
Find $BC$ .
21M.2.AHL.TZ2.11b:
Find the value of $k$ that satisfies the requirements of Pedro’s design.

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Topic 5 —Calculus