Approximation of decimal places, significant figures; Upper & Lower bounds, Percentage error, estimation: notes and practice questions

A cartographer, Elara, is calculating the diagonal distance across a square region with side length 1 unit. This involves the value of $\sqrt{2}$. She uses the following expression to approximate this value:

$1+\frac{1}{2+\frac{1}{2+\frac{1}{2}}}$

Calculate Elara's approximation, correct to four decimal places.

Calculate the percentage error in using Elara's four decimal place approximation of $\sqrt{2}$, compared to the exact value of $\sqrt{2}$ in your calculator.

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 manufacturing company purchases a new specialized machine. The machine's value depreciates exponentially over time. The value of the machine, $V$, in thousands of dollars, $t$ years after its purchase, is modelled by the function $V(t) = A e^{-kt}$, for $t \ge 0$.

The initial value of the machine was 150 thousand dollars. After 3 years, its value had decreased by 40%.

(a) Find the value of $k$.

(b) Calculate the value of the machine after 5 years and 6 months, giving your answer to two decimal places.

(c) The company believes that, according to this model, the machine will always have some residual value, however small.

State a mathematical reason why the company might believe this.

(d) Write down one possible limitation of the domain of the model.

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.

The diagram below shows a hot air balloon hovering at point H, $520$m vertically above a landing pad.

Point A is the point on the ground, directly below the hot air balloon.

An observer starts walking at a constant speed from point C towards point A. From point C, the observer looks upward at the hot air balloon at an angle of elevation of $30\degree$. After $20$ minutes, the observer is at point B and observes the same hot air balloon at an angle of elevation of $50\degree$.

Write down the size of the angle of depression from H to C.

Find the horizontal distance from A to C.

Calculate the distance the observer walked from C to B.

Determine the observer's average speed, in metres per hour.

(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.

A team of engineers is monitoring the flow of water into a large storage tank during a heavy rainfall. Let $F$ be the flow rate, in litres per hour, and $t$ be the time in hours since the monitoring began.

When $F$ is plotted against $t$, the total volume of water collected in the tank is represented by the area between the graph and the horizontal $t$ -axis.

The flow rate, $F$, is measured over the course of three hours. The results are shown in the following table.

(a) Use the trapezoidal rule with an interval width of 0.5 hours to estimate the total volume of water collected in the tank during these three hours.

(b) The actual volume of water collected during these three hours was 340 litres.

Find the percentage error of the estimate found in part (a).

A specialized chemical reactor is designed to produce a certain compound. The rate of change of the concentration of a key reactant, $C$, with respect to time, $t$, is modeled by the differential equation $\frac{dC}{dt} = \frac{t^2}{C}$.

At $t = 1$ minute, the concentration $C$ is $2$ mol/L.

(a) Use Euler's method with a step size of $0.1$ to find approximate values of $C$ when $t = 1.1$, $1.2$, and $1.3$ minutes. Give your answers to three decimal places.

(b) Solve the differential equation $\frac{dC}{dt} = \frac{t^2}{C}$ to find the exact concentration $C$ as a function of $t$. Hence, find the absolute errors for each of your approximations in part (a). Give your exact values to five decimal places and absolute errors to five decimal places.

A biologist is studying the effect of a new toxin on a bacterial culture. The number of active bacterial cells, $N$ (in thousands), remaining after exposure to a toxin dose $D$ (in mg/L) follows the relationship:

$\log_{10}N = k - D$, for some constant $k \in \mathbb{R}$.

In an experiment, it was observed that when the toxin dose was 2 mg/L, there were 1000 thousand (i.e., 1,000,000) active cells remaining.

(a) Find the value of $k$.

The relationship for this bacterial culture can also be written in the form $N = \frac{b}{10^D}$.

(b) Find the value of $b$.

(c) Given that the toxin dose $D$ is between 0.5 mg/L and 4.0 mg/L (i.e., $0.5 < D < 4.0$ ), find the range for $N$.

The effectiveness score, $S$, of a new antidote is inversely proportional to the number of active cells, $N$, remaining after toxin exposure, such that $S = \frac{1}{N}$.

(d) A new antidote was tested on a culture exposed to a dose of 4.5 mg/L. Calculate the effectiveness score, $S$, for this antidote. Give your answer to 3 significant figures.

A microbiologist is studying the interaction between two competing species of bacteria, species A and species B, in a controlled environment. Let $A$ represent the population of species A (measured in millions) and $B$ represent the population of species B (measured in millions). Let $t$ represent time, in hours.

The interaction can be modelled by the coupled differential equations:

$\frac{dA}{dt} = A(1 - B)$

$\frac{dB}{dt} = B(A - 3)$

State the two equilibrium points for this model.

Initially, there are $4$ million bacteria of species A and $0.5$ million bacteria of species B. Use the Euler method with a step size of $0.2$ hours to estimate the population of species A and species B after $1$ hour (to the nearest integer). Show the intermediate values that are obtained in the working, in the format of a table.

Suggest whether stating the population of bacteria to the nearest integer is a valid level of accuracy when using Euler's method in part (b).

The height of a diver above the water surface after jumping from a diving board is modelled by the function

$h(t) = -4.9t^2 + 5t + 10$

where $h(t)$ is the height in metres and $t$ is the time in seconds after the diver leaves the board.

(a) Write down the height of the diving board above the water surface.

(b) Find the value of $t$ when the diver enters the water. Give your answer to three significant figures.

(c) State an appropriate domain for $t$ in this model.

A construction company, 'BuildFast', is undertaking a large infrastructure project. The project is divided into 12 distinct phases.

Two teams, Team Alpha and Team Beta, are assigned to complete the project, working in parallel on different aspects.

Team Alpha's efficiency: The first phase takes them $3$ minutes $30$ seconds. Due to a new process, the time Team Alpha takes to complete each subsequent phase is $5$ seconds less than the previous phase.

Team Beta's efficiency: The first phase also takes them $3$ minutes $30$ seconds. Due to a continuous learning curve, the time Team Beta takes to complete each subsequent phase is $0.95$ times the time they took for the previous phase.

(a) (i) State the time Team Alpha takes to complete the third phase of the project.

(a) (ii) Show that Team Beta takes approximately $3$ minutes $10$ seconds (to the nearest second) to complete the third phase of the project.

(b) Both teams complete all $12$ phases of the project. Show that Team Beta completes the entire project faster than Team Alpha.

(c) Hence, state the value of the time difference, correct to the nearest second, between their total project completion times.

A colony of bacteria is growing in a nutrient solution. The rate of change of the population, $P$, with respect to time, $t$ (in hours), is modelled by the differential equation $\frac{dP}{dt} = P \cos t (e^{-\sin t})$. At time $t = 0$, the population is $P = 10$.

(a) By using Euler's method with a step length of 0.1, find an approximate value for the population when $t = 0.3$. Give your answer to three significant figures.

(b) By solving the differential equation, find the percentage error in your approximation for the population when $t = 0.3$. Give your answer to three significant figures.

A new electronics company, 'TechFlow', launched its flagship product. Let $n$ be the number of years since the product's launch. The sales team recorded the following data for the first two years.

Calculate the percentage increase in units sold from the first year to the second year.

It is assumed that the number of units sold each year will follow a geometric sequence, $u_n$.

Write down the common ratio of the sequence.

Find an expression for $u_n$.

Find the number of units TechFlow expects to sell when $n = 15$. Express your answer to the nearest integer.

In the first year, TechFlow's production facility had a capacity of $5500$ units. The company plans to increase its production capacity by $400$ units every year.

Let $v_n$ represent the production capacity of the facility in year $n$.

Write down an expression for $v_n$.

For the first $12$ years, TechFlow ensures that all units produced are sold. Each unit sold generates a profit of $30.

Calculate the total profit generated from units sold in the first $12$ years.

When $n = k$, the number of units demanded (sales) will, for the first time, exceed the production capacity.

Find $k$.

State whether, for all $n > k$, TechFlow will consistently have sales exceeding its production capacity.

Justify your answer.

A car enthusiast, Mr. Henderson, purchases two classic cars for his collection. The first is a popular sedan, initially costing $28000, which is expected to depreciate at a rate of 8$75000, and is expected to depreciate at a rate of 15% per year.

(a) Estimate the value of the popular sedan after 4 years.

(b) Find the number of years, $k$, after which both cars will have the same estimated value. Give your answer to three significant figures.

(c) Comment on the validity of your answer to part (b).

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 monument features a prominent pyramidal cap. The length of the slant edge from the apex, A, to any corner of its square base is measured as 2.8 m, accurate to the nearest tenth of a meter. The side length of the square base is exactly 3.2 m. Let C be a corner of the base.

Write down the upper bound and lower bound for the possible lengths of edge AC.

Let H be the midpoint of one of the base edges. Determine the upper bound and lower bound for AH, the slant height of the pyramid's triangular faces.

For structural stability, the angle between the slant height (AH) and the base of the pyramid must be less than 35°. Show whether this monument's pyramidal cap meets this stability requirement. Justify your answer.

A tech company launches a new social media app. The number of active users, $U$, can be modelled by the function

$U(t) = 800 \times k^t, t\ge 0$,

where $t$ is the number of hours since the app was launched, and $k$ is a positive constant.

Write down the value of $U(0)$.

Interpret what this value means in this context.

4 hours after the app was launched, the number of active users was 4050.

Find the value of $k$.

Find the number of active users 2 hours and 15 minutes after the app was launched.

A competitor launches a similar app, whose user base, $U_2$, can be modelled by the function

$U_2(t) = 2500 \times 1.2^t, t\ge 0$,

where $t$ is the number of hours since both apps were launched.

Find the value of $t$ when the number of users for both apps is equal.

It takes $H$ hours and $m$ minutes for the number of users of the first app to reach 15000.

Find the value of $H$ and $m$, giving $m$ as an integer.

Each user of the first app requires $1.5 \times 10^{-2}$ MB of server storage. The total available server capacity is $3.0 \times 10^5$ MB.

Determine how long it would take for the app's user base to exceed the server capacity.

The cross-section of a proposed tunnel entrance is modelled by a curve. The heights of the tunnel are measured at horizontal intervals and are given in the table below. All measurements are in cm.

Horizontal distance, x (cm)

0

10

20

30

40

50

60

Vertical distance, y (cm)

0

1.875

6.0

10.125

12.0

9.375

0

(a) Use the trapezoidal rule with $h = 10$ to find an approximation for the cross-sectional area of the tunnel entrance.

(b)

It is given that the equation of the curve is $y = 0.0225x^2 - 0.000375x^3$, for $0 \le x \le 60$.

(i) Write down an integral to find the exact cross-sectional area.

(ii) Calculate the value of the cross-sectional area to two decimal places.

(c) Find the percentage error in the area found using the trapezoidal rule. Give your answer to two decimal places.

A landscape architect is designing a new public park. The northern boundary of the park is modelled by the function $g(x)$, and other boundaries are straight lines. The plan view of the park is shown in the following diagram, where both axes represent distance and are measured in metres.

The function $g(x)$ models the northern boundary of the park between points B and C and is given by

$g(x) = -\frac{1}{10}x^2 + 4x + 15$, for $0 \le x \le 40$.

(i) Find $g'(x)$.

(ii) Hence find the coordinates of the point on the park's northern boundary that is furthest north.

Point A has coordinates $(0, 0)$, point B has coordinates $(0, 15)$, point C has coordinates $(40, 15)$ and point D has coordinates $(40, 0)$.

(i) Write down the integral which can be used to find the area of the shaded region representing the park.

(ii) Find the area of the park.

(i) A landscaper uses the trapezoidal rule with 4 intervals to estimate the area of the park. Calculate this estimate.

(ii) Calculate the percentage error in the landscaper's estimate.

(iii) Suggest how the landscaper might be able to reduce the error whilst still using the trapezoidal rule.

A square-shaped meditation garden, PQRS, is to be built within the park. The side [PS] lies on the southern boundary (x-axis), and the side [QR] lies on the eastern boundary of the park ($x=40$). Point Q lies on the northern boundary curve $g(x)$.

(i) Find the x-coordinate of point P for the largest area of the meditation garden.

(ii) Find the largest area of the meditation garden.