The diagram shows an open water container in the  shape of a triangular prism of length $y cm$.

The vertical cross-section is an isosceles triangle with sides $5x cm$.$5x cm$ and $6x cm$.

The water container is made from $500c{m}^2$ of sheet metal and has a volume of $V c{m}^3$.

Show that $V=600x-\frac{144}{5}{x}^3$.

 

The diagram shows an open water container in the  shape of a triangular prism of length $y cm$.

The vertical cross-section is an isosceles triangle with sides $5x cm$.$5x cm$ and $6x cm$.

The water container is made from $500c{m}^2$ of sheet metal and has a volume of $V c{m}^3$.

Value of $x$ for which $V$ has a stationary value is $\frac{5\sqrt{a}}{6}$. Find $a$.

The diagram shows an open water container in the  shape of a triangular prism of length $y cm$.

The vertical cross-section is an isosceles triangle with sides $5x cm$.$5x cm$ and $6x cm$.

The water container is made from $500c{m}^2$ of sheet metal and has a volume of $V c{m}^3$.

Find the value of $x$ for which $V$ has a stationary value.

Show that $V$ has a maximum at $x$.

The diagram shows a solid formed by joining a hemisphere of radius $r$ $cm$ to a cylinder of radius $r cm$ and height $h cm$. the surface area of the solid is $320\mathrm{\pi }c{m}^2$ and the volume is $V c{m}^3$.

Express $h$ in terms of $r$.

The diagram shows a solid formed by joining a hemisphere of radius $r$ $cm$ to a cylinder of radius $r cm$ and height $h cm$. the surface area of the solid is $320\mathrm{\pi }c{m}^2$ and the volume is $V c{m}^3$.

Show that $V=160\mathrm{\pi r}-\frac{5}{6}{\mathrm{\pi r}}^3$.

The diagram shows a solid formed by joining a hemisphere of radius $r$ $cm$ to a cylinder of radius $r cm$ and height $h cm$. the surface area of the solid is $320\mathrm{\pi }c{m}^2$ and the volume is $V c{m}^3$.

Find the exact value of $r$ such that $V$ is a maximum.

The diagram shows a right circular cone of base radius $r cm$ and height $h cm$ cut from a solid sphere of radius $10 cm$. the volume of the cone is $V c{m}^3$.

Express $r$ in terms of $h$.

The diagram shows a right circular cone of base radius $r cm$ and height $h cm$ cut from a solid sphere of radius $10 cm$. the volume of the cone is $V c{m}^3$.

Show that $V=\frac{1}{3}\mathrm{\pi }{h}^2\left(20-h\right)$.

The diagram shows a right circular cone of base radius $r cm$ and height $h cm$ cut from a solid sphere of radius $10 cm$. the volume of the cone is $V c{m}^3$.

Find the value for $h$ for which there is a stationary value of $V$.