Sue Pemberton Solutions for Chapter: Further Differentiation, Exercise 5: EXERCISE 8C

$PQRS$ is a rectangle with base length $2p$ units and area $A$ units². The points $P$ and $Q$ lie on the $x$-axis and the points $R$ and $S$ lie on the curve $y=9-x^2$. Find the value of $p$ for which $A$ has a stationary value.

$PQRS$ is a rectangle with base length $2p$ units and area $A$ units². The points $P$ and $Q$ lie on the $x$-axis and the points $R$ and $S$ lie on the curve $y=9-x^2$. Find the value of $p$ for which $A$ has a stationary value. Find this stationary value and determine its nature.

The diagram shows a $24cm$ by $15cm$ sheet of metal with a square of side $xcm$ removed from each corner. The metal is then folded to make an open rectangular box of depth $xcm$ and volume $Vc{m}^3$. Show that $V=4x^3-78x^2+360x$.

The diagram shows a $24cm$ by $15cm$ sheet of metal with a square of side $xcm$ removed from each corner. The metal is then folded to make an open rectangular box of depth $xcm$ and volume $Vc{m}^3$. Find the stationary value of $V$ and the value of $x$ for which this occurs.

The diagram shows a $24cm$ by $15cm$ sheet of metal with a square of side $xcm$ removed from each corner. The metal is then folded to make an open rectangular box of depth $xcm$ and volume $Vc{m}^3$. Find the stationary value of $V$ and the value of $x$ for which this occurs. Determine the nature of this stationary value.

The volume of the solid cuboid shown in the diagram is $576c{m}^3$ and the surface area is $Ac{m}^2$.  Express $y$ in terms of $x$.

The volume of the solid cuboid shown in the diagram is $576c{m}^3$ and the surface area is $Ac{m}^2$.  Show that $A=4x^2+\frac{1728}{x}$

The volume of the solid cuboid shown in the diagram is $576c{m}^3$ and the surface area is $Ac{m}^2$. Find the minimum value of $A$ and state the dimensions of the cuboid for which this occurs.