# Applications of Differentiation

###### 5 Essential Questions

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##### 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. 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.)



##### Suggested Handwritten and Video Solutions  ##### 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.

##### Suggested Handwritten and Video Solutions
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