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###### A-Level H2 Math | 5 Essential Questions

# Applications of Differentiation

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**2020 NYJC CT2 P2 Q3**

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**2019 DHS Promo Q12**

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**2007 AJC P1 Q13 (b)**

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

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

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Differentiation in calculus is a key concept used to determine instantaneous rates of change. Instantaneous rate of change is the measure of how quickly the new value is affecting the old value. Differentiation gives us the ability to isolate and measure values between two variables instead of just their sum or average.

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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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[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.

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

[2]

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

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

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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 $

[3]

(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]

- (i)
- (ii)

- (i)
- (ii)

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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.

[5]

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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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