Volume & surface area of 3D objects: notes and practice questions

A company produces artisanal candles. One of their products is a large conical candle with a base radius of 6 cm and a height of 10 cm.

(a) Find the volume of the conical candle, giving your answer in terms of $\pi$.

The wax from one of these conical candles is melted down and recast to create 20 identical cylindrical tea light candles, each with a height of 5 cm.

(b) Find the radius of a tea light candle.

Consider the function $f$ defined by $f(x) = x^3 - 6x^2 + 8x$.

Find the $x$-intercepts of the graph of $y=f(x)$.

The graph of $y=f(x)$ for $0 \le x \le 4$ is shown below. The graph encloses two regions with the $x$-axis, shaded in the diagram.

Find the total area of the shaded regions.

The total surface area of a closed right cylinder is 8, equal to the total shaded area found in part (b). The cylinder has a height of $\frac{4-\pi}{\pi}$.

Find the radius, $r$, of the cylinder.

Hence, find the volume of the cylinder.

Two spacecraft, S1 and S2, travel along straight paths, represented by the lines $L_1$ and $L_2$ respectively. The paths of the spacecraft intersect at a docking station D. A probe is located at a point P on the path of $L_2$. This is shown in the following diagram.

The direction vector of $L_1$ is $\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}$. The vector $\vec{\text{DP}}$ is given by $\begin{pmatrix} k \\ 4 \\ 0 \end{pmatrix}$, where $k \ge 0$.

The acute angle between the paths $L_1$ and $L_2$ is $\theta$, where $\cos\theta = \frac{1}{3}$.

(a) Show that $2k+4 = \sqrt{k^2+16}$.

(b) Find the value of $k$.

(c) Hence, find the shortest distance from the probe at P to the path $L_1$.

The paths $L_1$ and $L_2$ lie on a plane, $\Pi$.

(d) Find a vector normal to the plane $\Pi$.

A satellite dish is modelled as a right circular cone with its vertex at V. The base of the cone lies in the plane $\Pi$ and is centred at P. The path $L_1$ is tangent to the circular base of the cone. The volume of the cone is $\frac{128\pi\sqrt{2}}{9}$ cubic units. The position vector of P is $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.

(e) Find the two possible position vectors for V.

A scientist is studying a spherical microbe. The radius of the microbe is measured to be $3 \times 10^{-5}$ cm.

(a) Write down the diameter of the microbe.

(b) The volume of the microbe can be expressed in the form $\pi(a \times 10^k) \text{ cm}^3$ where $1 \leq a < 10$ and $k \in \mathbb{Z}$.

Find the value of $a$ and the value of $k$.

A rectangle is inscribed in an ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$. The sides of the rectangle are parallel to the coordinate axes. The vertices of the rectangle are located at $(\pm x, \pm y)$, where $x > 0$ and $y > 0$.

(a) Show that the area of the rectangle, $A$, can be expressed as $A = \frac{12x}{5}\sqrt{25-x^2}$.

(b) Show that $\frac{dA}{dx} = \frac{12(25-2x^2)}{5\sqrt{25-x^2}}$.

(c) Hence, find the exact dimensions of the rectangle with the maximum possible area.

A kite PQRS is shown on the following set of axes.

The kite has vertices Q(1, 5) and S(7, -1) and is symmetrical about the diagonal [PR].

(i) Write down the coordinates of the midpoint of [QS].

(ii) Hence, find the equation of the line containing the diagonal [PR].

(b) Given that vertex P lies on the y-axis and the x-coordinate of R is 6, find the area of the kite PQRS.

A hollow pipe is manufactured by removing a smaller cylinder of radius $r$ from the centre of a larger cylinder of radius $3r$. Both cylinders have the same height, $h$. This is shown in the following diagram.

All lengths are measured in centimetres.

The total surface area of the hollow pipe, in cm$^2$, is given by $S$.

(a) Show that $S = 16\pi r^2 + 8\pi rh$.

The total surface area of the hollow pipe is $128\pi \text{ cm}^2$.

(b) Show that the volume of the pipe, $V$, is given by $V = 128\pi r - 16\pi r^3$.

(c) Find an expression for $\frac{dV}{dr}$.

(d) The hollow pipe has its maximum volume when $r = k\sqrt{6}$, where $k \in \mathbb{Q}^+$. Find the value of $k$.

(e) Hence, find this maximum volume, giving your answer in the form $m\pi\sqrt{6}$, where $m \in \mathbb{Q}^+$.

An engineer is designing an open-top rectangular container with a square base. The container must have a volume of $4\text{ m}^3$.

Let the side length of the square base be $x$ metres and the height of the container be $h$ metres.

Show that the total surface area, $S\text{ m}^2$, of the material used for the container is given by $S = x^2 + \frac{16}{x}$.

Find an expression for $\frac{dS}{dx}$.

Hence, find the exact value of $x$ for which the surface area is a local minimum or maximum.

Find an expression for $\frac{d^2S}{dx^2}$.

Use the second derivative of $S$ to justify that $S$ is a minimum when $x = 2$.

Find the minimum surface area of the container.

The function $f$ is defined by $f(x) = e^x \sin(x)$, for $x \in [0, \pi]$.

(a) Show that the curve $y=f(x)$ has only one point of inflection in its domain, and determine its coordinates.

(b) Find the equations of the tangent and the normal to the curve at the point where $x = \pi$.

(c) Calculate the area of the triangle formed by this tangent, this normal, and the $y$-axis.

A packaging company is designing a new cylindrical can. The can must have a fixed volume of $54\pi$ cm$^3$. The company wants to minimize the amount of material used, which corresponds to minimizing the total surface area of the can.

Let the radius of the can be $r$ cm and its height be $h$ cm.

Show that the total surface area, $S$ cm$^2$, of the can is given by $S = 2\pi r^2 + \frac{108\pi}{r}$.

The total surface area of the can has a local minimum value when $r = a$.

(i) Find an expression for $\frac{dS}{dr}$.

(ii) Hence, find the exact value of $a$.

(i) Find an expression for $\frac{d^2S}{dr^2}$.

(ii) Use the second derivative of $S$ to justify that $S$ is a minimum when $r = a$.

(iii) Find the minimum surface area of the can.

(a) A decorative rocket toy is designed with a cylindrical body topped by a cone. The cylindrical part has a radius of $3$ cm and a height of $8$ cm. The conical part also has a radius of $3$ cm and a height of $4$ cm.

Calculate the total surface area of the rocket toy, giving your answer to one decimal place.

(b) Calculate the total volume of the rocket toy, giving your answer to one decimal place.

(c) The rocket toy is to be placed inside a rectangular prism-shaped display case with internal dimensions $6.5$ cm by $6.5$ cm by $12.5$ cm. Determine the percentage of the volume of the display case that is occupied by the rocket toy. Give your answer to one decimal place.

(a) A confectioner prepares a batch of $120$ spherical chocolate truffles, each with a diameter of $4 \text{ cm}$.

Calculate the total volume of all $120$ truffles, giving your answer to the nearest whole number.

(b) The truffles are melted down and poured into a rectangular mold to form a solid chocolate bar. The base of the mold has dimensions $30 \text{ cm}$ by $20 \text{ cm}$.

Calculate the height of the chocolate bar, correct to three significant figures.

A decorative perfume bottle is designed with a cylindrical base and a conical top. The cylindrical part has a radius of $2.5 \text{ cm}$ and a height of $8 \text{ cm}$. The conical part has the same radius as the cylinder and a height of $4 \text{ cm}$.

(a) Calculate the total volume of the perfume bottle.

(b) The external surface of the bottle needs to be coated with a protective layer. Calculate the total external surface area of the bottle, assuming the base of the cylinder is flat and also coated, but the interface between the cylinder and cone is not coated.

(a) A deep-sea research capsule is designed in the shape of a cylinder of length $12 \text{ m}$ and diameter $4 \text{ m}$, with a hemispherical dome welded to each end.

Calculate the total internal volume of the research capsule.

(b) The external surface of the capsule needs to be coated with a protective layer.

Determine the total external surface area of the research capsule.

A decorative garden pillar is constructed from a solid cylindrical base topped by a solid hemisphere. The radius of both the cylinder and the hemisphere is $4$ cm, and the height of the cylindrical part is $10$ cm.

(a) Calculate the total volume of the decorative pillar.

(b) Calculate the total exterior surface area of the decorative pillar.

(a) A glamping tent is designed in the shape of a rectangular-based pyramid. The base of the tent measures $8$ m by $6$ m, and the vertical height of the tent from the centre of the base to the apex is $4$ m.

Calculate the volume of the tent.

(b) The tent is advertised to accommodate a certain number of people, with each person requiring $3.2 \text{ m}^3$ of air space. Using your answer from part (a), calculate the maximum number of people the tent can accommodate.

(c) Show that the length of the longest sloping edge of the tent is $6.40$ m, correct to 3 significant figures.

(d) Find the angle at the apex between two adjacent longest sloping edges, specifically the angle formed by the two sloping edges that meet at the apex and span the $6$ m width of the base. Give your answer in degrees, correct to one decimal place.

(e) Calculate the total surface area of the tent fabric, excluding the base. Give your answer correct to one decimal place.