###### A-Level H2 Math

# Applications of Differentiation

###### 5 Essential Questions

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- Q2
- Q3
- Q4
- Q5

**2020 NYJC CT2 P2 Q3**

The diagram below shows a city map of two towns, $A$ and $B$ separated by a river. A bridge is to be built between the two towns, which are on opposite sides of a straight river of uniform width $r$km, and the two towns are $p$ km apart measured along the riverbank. Town $A$ is $1$km from the riverbank, and Town $B$ is $b$km away from riverbank.

A bridge is to be built perpendicular to the riverbank at a distance of $x$km from Town $B$, measured along the riverbank, allowing traffic to flow between the two towns.

Â

Find the distance $x$, in terms of $b$ and $p$, such that the distance of travel between Town $A$ and Town $B$ can be minimised if $b>1$. (It is not necessary to verify that the distance is minimum.)

[9]

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**2019 DHS Promo Q12**

[A circular cone with base radius $r$, vertical height $h$ and slant height $l$, has curved surfaced area $\pi rl$ and volume $\frac{1}{3}\pi r{{h}^{2}}$.]

A capsule made of metal sheet of fixed volume $p$cm$^{3}$ is made up of three parts.

- The top is modelled by the curved surface of a circular cone of radius $r$cm. The ratio of its height to its base radius is $4:3$.
- The body is modelled by the curved surface of a cylinder of radius $r$cm and height $H$cm.
- The base is modelled by a circular disc of radius $r$cm.

The cost of making the body of the capsule is $k$ per cm$^{2}$, while that of the top and the base of the capsule is $2k$ per cm$^{2}$, where $k$ is a constant. The total cost of making the capsule is $ $C$.

Assume the metal sheet is made of negligible thickness.

(i)

Show that $H=\frac{p}{\pi {{r}^{2}}}-\frac{4r}{9}$.

[2]

(i) Show that $H=\frac{p}{\pi {{r}^{2}}}-\frac{4r}{9}$.

[2]

(ii)

Express $C$ in the form $\frac{A}{r}+B{{r}^{2}}$, where $A$ and $B$ are expressions in terms of $k$ and $p$. Use differentiation to show that $C$ has a minimum as $r$ varies.

[8]

(ii) Express $C$ in the form $\frac{A}{r}+B{{r}^{2}}$, where $A$ and $B$ are expressions in terms of $k$ and $p$. Use differentiation to show that $C$ has a minimum as $r$ varies.

[8]

(iii)

Hence determine the ratio of $H$ to $r$ when $C$ is a minimum.

[2]

(iii) Hence determine the ratio of $H$ to $r$ when $C$ is a minimum.

[2]

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- (i)
- (ii) (a)
- (ii) (b)
- (iii)

- (i)
- (ii) (a)
- (ii) (b)
- (iii)

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**2007 AJC P1 Q13 (b)**

Water is poured at a constant rate of $20\,\text{c}{{\text{m}}^{3}}$ per second into a cup which is shaped like a truncated cone as shown in the figure. The upper and lower radii of the cup are $4\,\text{cm}$ and $2\,\text{cm}$ respectively. The height of the cup is $\text{6}\,\text{cm}$.

(i)

Show that the volume of water inside the cup, $V$ is related the height of the water level, $h$ through the equation

$V=\frac{\pi }{27}{{(h=6)}^{3}}-8\pi $

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(i) Show that the volume of water inside the cup, $V$ is related the height of the water level, $h$ through the equation

$V\,=\,\frac{\pi }{27}{{(h=6)}^{3}}-8\pi $

[3]

(ii)

How fast will the water level be rising when $h$ is $\text{3}\,\text{cm}$?

Express your answer in exact form.

[3]

(ii) How fast will the water level be rising when $h$ is $\text{3}\,\text{cm}$?

Express your answer in exact form.

[3]

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- (i)
- (ii)

- (i)
- (ii)

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**2018 MJC P1 Q9 (ii) Modified**

A manufacturer produces closed hollow cans. The top part is a hemisphere made of tin and the bottom part is a cylinder made of aluminium of cross-sectional radius $r$ cm and $h$ cm. There is no material between the cylinder and the hemisphere so that any fluid can move freely within the container. At the beginning of an experiment, a similar-shaped can of dimensions $r=4$and $h=10$, is filled to its capacity with water. Due to a hole at its base, water is leaking at a constant rate of 2 $\text{c}{{\text{m}}^{3}}{{\text{s}}^{-1}}$ when the can is standing upright. Find the exact rate at which the height of the water is decreasing 80 seconds after the start of the experiment.

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##### The Maximum Area of An Isosceles Triangle Inscribed In A Circle

An isosceles triangle is inscribed in a circle of radius $r$. Find the maximum area of this triangle in terms of $r$.

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- I
- II

- I
- II

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**Download Applications of Differentiation Worksheet**

**H2 Math Question Bank**

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