Practice section 3 of 8

The tangent at $Q(t,e^t)$ on $y=e^x$ meets the x-axis at $T$. The normal at $Q$ meets the x-axis at $N$. Express the area of triangle $QNT$ in terms of $t$.

Example 11 (Self-Reading): The curve $C$ has equation $\sin(x+y)=\sin x+\cos y$. Its normal at $P(\pi/4,\pi/2)$ meets the x- and y-axes at $S$ and $T$ respectively. Find the exact area of triangle $OST$, where $O$ is the origin, in the form $\dfrac{\pi^2}{64}(a+b\sqrt c)$, where $a,b,c$ are positive integers to be determined. [7]

Air is pumped into a spherical balloon at $10\text{ cm}^3/\text{s}$. Find the exact rate of change of its radius when the radius is $5\text{ cm}$. Given $V=\dfrac43\pi r^3$.

A vessel containing water has the shape of an inverted right circular cone with base radius $2\text{ m}$ and height $5\text{ m}$. Water flows from its vertex at $0.2\text{ m}^3/\text{s}$. Find the exact rate at which the water level is dropping when the water depth is $4\text{ m}$.

A closed thin-metal container is a cylinder with a hemispherical cap at one end. Find the minimum area of metal required for a volume of $100\text{ cm}^3$. Given sphere volume $\frac43\pi r^3$ and surface area $4\pi r^2$.

Example 15 (Self-Reading): An isosceles triangle $ABC$, with $AB=AC$, is inscribed in a fixed circle of radius $a$, with $\angle BAC=2\theta$. Express its area in terms of $\theta$, and hence show that the area is maximum when the triangle is equilateral.