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.

Find the value of $a$.

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 $\frac{2\pi }{3}$.

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.