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

A company manufactures decorative glass paperweights in the shape of a solid hemisphere. The radius of each paperweight is 8 cm.

(a) Calculate the total surface area of one such paperweight.

Each paperweight is coated with a protective sealant. It is known that 1 mL of the sealant covers an area of 150 cm$^2$.

(b) Calculate the volume of sealant required to coat one paperweight.

A scientist is preparing a cylindrical container for an experiment. She measures the radius of the base, $r$, to be 5.0 cm and the height, $h$, to be 12 cm.

Calculate the volume of the cylindrical container using these measurements. Give your answer to three significant figures.

It is known that the measurements are accurate to the number of significant figures given.

Find the lower bound and upper bound of the volume of the cylindrical container. Give your answers to three significant figures.

Find, with justification, the largest possible percentage error if the answer to part (a) is recorded as the volume of the cylindrical container.

(a) A satellite dish is supported by three anchor points A, B, and C on a mast. The coordinates of these points, relative to a fixed origin, are A(1, 2, 0), B(7, 2, 0), and C(4, 6, 1), where distances are measured in metres.

(i) Find the vector $\overrightarrow{\text{CA}}$.

(ii) Find the vector $\overrightarrow{\text{CB}}$.

(b) Calculate $\overrightarrow{\text{CA}} \times \overrightarrow{\text{CB}}$.

(c) Hence, determine the area of the triangular support surface formed by points A, B, and C.

A rectangular prism with length 10 cm, width 8 cm, and height 6 cm is used as a base for a sculpture. One corner, vertex H, is cut off by a flat plane passing through points P, Q, and R. Point P is on edge HE such that HP = 3 cm. Point Q is on edge HG such that HQ = 4 cm. Point R is on edge HD such that HR = 3 cm.

(a) Calculate the volume of the remaining part of the prism.

(b) Calculate the total surface area of the remaining part of the prism.

A landscape architect is designing a section of a park. The boundary of a planned pathway can be modelled by the line $y = -x + 8$ and a decorative flower bed by the curve $y = 0.5x^2 - 4x + 10.5$. These two features intersect at points $(1, 7)$ and $(5, 3)$, as shown in the following diagrams.

In diagram 1, the region enclosed by the line $y = -x + 8$, $x = 1$, $x = 5$ and the $x$-axis has been shaded.

Calculate the area of the shaded region in diagram 1.

In diagram 2, the region enclosed by the curve $y = 0.5x^2 - 4x + 10.5$, and the lines $x = 1$, $x = 5$ and the $x$-axis has been shaded.

Write down an integral for the area of the shaded region in diagram 2.

Calculate the area of this region.

Hence, determine the area enclosed between the line $y = -x + 8$ and the curve $y = 0.5x^2 - 4x + 10.5$.

(a) A cylindrical grain silo with a radius of $2.5$ m and an initial grain height of $10$ m is being emptied. The rate at which the grain is removed from the silo is proportional to the square root of the height of the grain.

Using $V$ for the volume of grain in the silo and $h$ for the height of the grain, write down a differential equation relating these variables.

(b) Apply the chain rule and the formula for the volume of a cylinder to show that $\frac{dV}{dt} = 6.25\pi \frac{dh}{dt}$. Hence, write your answer to part (a) in terms of $h$ and $t$.

(c) After $30$ minutes, the height of the grain has dropped to $6.25$ m. Predict how long it will take for the silo to be completely empty from the initial height of $10$ m.

The ancient civilization of Eldoria constructed monumental structures using a unit of measurement called the Ancient Eldorian Rod (AER).

Historians have determined that 1 AER is equivalent to 0.74 metres, rounded to two decimal places.

Write down the upper and lower bounds of 1 AER in metres.

The grand 'Tower of Eldoria' is modelled as a square-based pyramid with a base side length of 380 AER and a height of 250 AER. Assume these measurements are exact.

Find the minimum possible volume of the pyramid in cubic metres.

A company is designing an open-top storage container with a square base. The side length of the base is $x$ cm and the height is $h$ cm. The container needs to have a volume of $500$ cm$^3$.

Explain why $x^2 h = 500$.

Rearrange the equation in part (a) to make $h$ the subject.

Write down an expression for the surface area, $A$, of the open-top container.

Show that this can be written as

$A = x^2 + \frac{2000}{x}$

Plot the graph of $A = x^2 + \frac{2000}{x}$ for $x > 0$.

Find the minimum surface area and the value of $x$ when this occurs.

A spherical ice sculpture is melting in a gallery. Initially, the sculpture has a radius of 30 cm. This information is illustrated in the following diagram.

The gallery curator predicts that, as the ice sculpture melts, its radius will decrease at a constant rate of 0.5 cm per hour.

According to this model, find

(i) the radius of the ice sculpture, 10 hours after it begins melting.

(ii) the volume of the ice sculpture, 10 hours after it begins melting. Give your answer to one decimal place.

Let the function $V(t)$ represent the volume of the ice sculpture, $\text{cm}^3$, $t$ hours after it begins melting. $V(t)$ is given by

$V(t) = 113000 - 5000t + 150t^2 - 1.5t^3$, for $0 \le t \le 20$.

Find $V'(t)$.

Find the rate of change of the volume of the ice sculpture at $t = 10$ hours.

State one reason why the radius of the ice sculpture may not always decrease at a constant rate.

(a) A pharmaceutical company is designing a new cylindrical container for a special liquid. The container has a fixed volume $V = 500 \text{ cm}^3$.

The radius of the container is $r \text{ cm}$ and the height is $h \text{ cm}$.

Show that $\pi r^2 h = 500$.

(b) Find an expression for the total surface area $S$ of the container.

(c) Substitute an expression for $h$ (from part (a) ) into your expression for $S$ (from part (b) ) and hence show that $S = 2\pi r^2 + \frac{1000}{r}$.

(d) Find $\frac{dS}{dr}$.

(e) Find the minimum value of $S$ and the values of $r$ and $h$ when this occurs. Show that this value of $S$ is indeed a minimum.

(a) A spherical decorative orb has a radius of 4.5 cm. A craftsman plans to cover $\frac{2}{7}$ of its surface with intricate filigree work.

Calculate the volume of the orb that will NOT be covered by the filigree work.

(b) A conical party hat has a slant height of 15 cm and a base radius of 6 cm. The outside of the hat (excluding the base) is to be decorated with glitter.

Calculate the surface area of the hat that will be covered with glitter. Give your answer correct to the nearest cm$^2$.

A landscape architect is designing a large decorative fountain for a new public park. The outer structure of the fountain consists of a cylindrical base topped by a conical section. The inner part of the fountain, which holds the water, is a hollow space created by rotating a specific curve around the vertical axis.

The shape of the inner hollow is based on a transformation of the graph $y = -x^3$. The curve that defines the profile of the inner hollow is given by $y = 40 - 0.064x^3$. This transformation involves a vertical translation of $a$ units and a stretch parallel to the x-axis with a scale factor of $b$.

(a.i) Write down the value of $a$.

(a.ii) Find the value of $b$.

The cylindrical base of the fountain has a radius of $15$ m and a height of $5$ m. The conical section on top has the same base radius of $15$ m and a height of $8$ m. The inner hollow, described by the curve $y = 40 - 0.064x^3$, extends from the base of the fountain ($y=0$) up to the total height of the outer structure.

Find the volume of the solid material that makes up the fountain (i.e., the volume of the outer structure minus the volume of the inner hollow).

A conical tank is being filled with water. The tank has a height of 12 cm and a radius of 4 cm at its top. Water is poured into the tank such that the volume of water is increasing at a rate of 10 cm$^3$s$^{-1}$.

(a) Find the height of the water in the tank when its volume is 8$\pi$ cm$^3$.

(b) Hence or otherwise, find the rate of change of the height of the water at this instant.

A botanical garden is constructing a large planter box for exotic plants. The planter is shaped like an inverted frustum of a right pyramid, with a horizontal square top opening of side length $120$ metres and a smaller horizontal square base of side length $80$ metres.

The depth of the planter is $9$ metres.

Find the angle of inclination of the side walls of the planter to the horizontal.

(b) The point V is the theoretical vertex of the full pyramid from which the frustum is cut, and C is the centre of the square top opening.

(i) Find the total height of the pyramid from its theoretical vertex V to the centre of the top opening C.

(ii) Hence or otherwise, show that the volume of the planter is $91200 \text{ m}^3$.

The botanical garden orders $180000$ bags of a special soil mix, with each bag containing $0.5 \text{ m}^3$ of soil. Determine whether the ordered soil is sufficient to fill the planter.

To prevent soil erosion and water leakage, the entire interior surface of the planter (including the bottom base and the four side walls) needs to be lined with a waterproof membrane. Calculate the total area that needs to be lined.

(a) A sculptor is creating a metal art piece from a solid cylindrical block. The block has a radius of $5$ cm and a height of $12$ cm. A conical section is drilled out from the top center of the block. The conical hole has the same radius as the cylinder and a height of $9$ cm.

Calculate the remaining volume of the metal sculpture.

(b) Determine the total surface area of the finished sculpture.

(a) An architect is designing a modular facade for a building. Each module is based on a fundamental rectangular panel, $P$. The vertices of panel $P$ are $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$.

Show that the area of the panel $P$ is $6$ square units.

(b) The design incorporates 30 elements, each obtained by transforming the panel $P$. These elements form a symmetric pattern, with the y-axis acting as a line of symmetry.

The transformation that produces each of the elements on the right side of the design can be represented by a matrix of the form

$M_k = \begin{pmatrix} (1-\frac{k}{15})\cos(k\times12\degree) & -(1-\frac{k}{15})\sin(k\times12\degree) \\ (1-\frac{k}{15})\sin(k\times12\degree) & (1-\frac{k}{15})\cos(k\times12\degree) \end{pmatrix}$

where $k = 0, 1, 2, ..., 14$.

(i) Find the matrix $M_0$. Give your answer in the form $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ where $a, b, c, d \in \mathbb{Q}$.

(ii) Hence find the coordinates of the image of the vertex $(3,2)$ after it is transformed by the matrix $M_0$.

(c) The matrix $M_k$ can be expressed as the product of a rotation matrix and an enlargement matrix.

(i) Write down, in terms of $k$, the rotation matrix.

(ii) Write down, in terms of $k$, the enlargement matrix.

(iii) Write down, in terms of $k$, the angle of the rotation.

(iv) Write down, in terms of $k$, the scale factor of the enlargement.

(d) Using your answer to part (c)(iv), or otherwise, find the determinant of the matrix $M_k$ in terms of $k$.

(e) Hence, or otherwise, find the total area of the elements in the whole design. Give your answer to three significant figures.

(f) Each element on the left side of the design can be obtained through a transformation of the panel $P$ by applying the matrix $N_k$, where $k = 0, 1, 2, ..., 14$.

Write down the matrix $N_k$ as a product of two matrices.

A company is designing heat sinks for a new line of high-performance microchips. They want to maximize the heat dissipation efficiency, which is directly related to the surface area to volume ratio ($SA:V$) of the heat sink material. A higher $SA:V$ ratio allows for faster heat transfer away from the chip.

(a) Calculate the surface area, volume, and surface area to volume ratio ($SA:V$) for each of the heat sink designs below. Give your answers to three significant figures where appropriate.

(i) Square Prisms

(ii) Cylinders

(b) Based on your calculations, state what conclusions can be drawn about the design principles for heat sinks to achieve a higher surface area to volume ratio ($SA:V$).

(a) A confectionery company is designing a new wrapper for a chocolate bar. The wrapper is in the shape of a rectangular prism with a square base of side length $x$ cm. The height, $h$ cm, is twice the side length of the base.

Write down an expression for $h$ in terms of $x$.

(b) The chocolate bar has a volume of $250$ cm$^3$.

Find the value of $x$ and $h$.

(c) Calculate the total external surface area of this rectangular prism wrapper.

(d) The company also considers a cylindrical wrapper with radius $r$ cm and height $H$ cm. This wrapper must also hold $250$ cm$^3$ of chocolate.

Find an expression for the height, $H$, of the cylindrical wrapper in terms of $r$.

(e) Let the total external surface area of the cylindrical wrapper be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{dA}{dr}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ for the cylindrical wrapper.

(h) To account for manufacturing waste and overlap, an additional $12\%$ of the calculated surface area is required for the rectangular prism wrapper, and $20\%$ for the cylindrical wrapper.

Determine which wrapper design the company should choose to minimize material usage. Justify your answer.

A rectangular prism (cuboid) is used as a building block. It has one vertex at the origin O(0,0,0). Its dimensions are 6 units along the x-axis, 4 units along the y-axis, and 3 units along the z-axis. The vertices are given by O(0,0,0), A(6,0,0), B(6,4,0), C(0,4,0), D(0,0,3), E(6,0,3), F(6,4,3), and G(0,4,3).

Find the surface area of the cuboid.

Find the length of the space diagonal [OF].

The space diagonals [OF] and [CE] intersect at the point M.

Find the coordinates of M.

Find the acute angle between the diagonals [OF] and [CE] at their intersection point M.

(a) SweetTreats is designing new packaging for a line of artisanal chocolates. The initial design is a box in the shape of a cuboid with a square base of side length $L$ cm. Its height, $H$ cm, is twice the length of the base.

Write down an expression for $H$ in terms of $L$.

(b) The box is designed to hold $250$ cm$^3$ of chocolates.

Find the value of $L$ and $H$.

(c) Calculate the total external surface area of the box.

(d) To minimize the amount of material needed, SweetTreats is considering changing the shape to a cylinder with radius $r$ cm and height $h$ cm. The cylindrical container must also hold $250$ cm$^3$ of chocolates.

Find an expression for the height, $h$, of the container in terms of $r$.

(e) Let the total external surface area of the cylindrical container be $A$ cm$^2$.

Show that $A = 2\pi r^2 + \frac{500}{r}$.

(f) Find $\frac{\text{d}A}{\text{d}r}$.

(g.i) Hence or otherwise, find the value of $r$ that will minimize $A$.

(g.ii) Find the minimum value of $A$ needed for the cylinder.

(h.i) Find $\frac{\text{d}^2 A}{\text{d}r^2}$.

(h.ii) Hence determine whether the graph of $A$ is concave-up or concave-down for $r > 0$. Justify your answer.