Hence, determine the maximum volume of the container.

The following diagram shows the curve $\frac{x^2}{36}+\frac{{\left(y-4\right)}^2}{16}=1$, where $h\le y\le 4$.

The curve from point $\text{Q}$ to point $\text{B}$ is rotated $360°$ about the $y$-axis to form the interior surface of a bowl. The rectangle $\text{OPQR}$, of height $h \text{cm}$, is rotated $360°$ about the $y$-axis to form a solid base.

The bowl is assumed to have negligible thickness.

Given that the interior volume of the bowl is to be $285 {\text{cm}}^3$, determine the height of the base.

Let $R$ be the region enclosed by the graph of $f$, the $x$-axis and the lines $x=p$ and $x=b$. The region $R$ is rotated 360º about the $x$-axis. Find the volume of the solid formed.

Find the rate of change of $f$ at $\text{B}$.

When $t$ = 0, the volume of water in the container is 2.3 m³. It is known that the container is never completely full of water during the 4 hour period.

Find the minimum volume of empty space in the container during the 4 hour period.

Find the equation of the tangent to the graph of $f$ at $\text{A}$.

Find the value of $p$.

Find the volume of the container.

The following diagram shows part of the graph of $f$.

The region enclosed by the graph of $f$, the $x$-axis and the lines $x=-p$ and $x=p$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Write down the coordinates of $\text{A}$.

Find the value of $p$.

Find the volume of the solid formed when the region bounded by the curve, the $x$-axis for $x⩾0$ and the $y$-axis for $y⩾0$ is rotated through $2\pi$ about the $x$-axis.

Show that the volume, $V {\text{m}}^3$, of water in the container when it is filled to a height of $h$ metres is given by $V=\pi \left(\frac{1}{3}{h}^3+h\right)$.

State the equation of the horizontal asymptote on the graph of $y=f\left(x\right)$.

Sketch the graph of $y=f\left(x\right)$, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.

Hence find the area of the shaded region.

State the cross-sectional radius of the bowl at this point.

By sketching the graph of a suitable derivative of $f$, find where the cross-sectional radius of the bowl is decreasing most rapidly.

Use the model to find the volume of the barrel.

Find the value of $p$ and of $q$.

Determine the values of $p$, $q$ and $r$.

Find the coordinates of $\text{B}$.

The region $R$ is now rotated about the $y$-axis, through $2\pi$ radians, to form a solid.

By writing $\text{si}{\text{n}}^{3}y$ as $\left(1-\text{co}{\text{s}}^{2}y\right)\text{sin}y$, show that the volume of the solid formed is $\frac{2\pi }{3}$.

Find the volume of the solid formed when the shaded region is revolved $360°$ about the $x$-axis.

The empty barrel is being filled with water. The volume $V{\text{m}}^3$ of water in the barrel after $t$ minutes is given by $V=0.8(1-{\text{e}}^{-0.1t})$. How long will it take for the barrel to be half-full?

Find $\text{OA}$.

State the equation of the vertical asymptote on the graph of $y=f\left(x\right)$.

Use an algebraic method to determine whether $f$ is a self-inverse function.

Show that $\frac{3}{x+1}-\frac{1}{x-1}=\frac{2x-4}{x^2-1}$.

The region bounded by the $x$-axis, the curve $y=f\left(x\right)$, and the lines $x=5$ and $x=7$ is rotated through $2\mathrm{\pi }$ about the $x$-axis. Find the volume of the solid generated, giving your answer in the form $\mathrm{\pi }(a+b \ln 2)$, where $a, b \in \mathrm{\mathbb{Z}}$.

Find the area of $R$.

During the interval $p$ < $t$ < $q$, he volume of water in the container increases by $k$ m³. Find the value of $k$.

The area enclosed by the graph of $y=f\left(x\right)$ and the line $y=4$ can be expressed as $\text{ln}v$. Find the value of $v$.

Show that the volume of the solid formed is $\frac{15k^2\mathrm{\pi }}{34}$ cubic units.

Find $\text{BC}$.

Find the value of $k$ that satisfies the requirements of Pedro’s design.