Points $\text{A}$ and $\text{P}$ lie on opposite banks of a river, such that $\text{AP}$ is the shortest distance across the river. Point $\text{B}$ represents the centre of a city which is located on the riverbank. $\text{PB}=215 \text{km}$, $\text{AP}=65 \text{km}$ and $\text{A}\hat{\text{P}}\text{B}=90°$.

The following diagram shows this information.

A boat travels at an average speed of $42 {\text{km\hspace{0.05em}h}}^{-1}$. A bus travels along the straight road between $\text{P}$ and $\text{B}$ at an average speed of $84 {\text{km\hspace{0.05em}h}}^{-1}$.

Find the travel time, in hours, from $\text{A}$ to $\text{B}$ given that

There is a point $\text{D}$, which lies on the road from $\text{P}$ to $\text{B}$, such that $\text{BD}=x \text{km}$. The boat travels from $\text{A}$ to $\text{D}$, and the bus travels from $\text{D}$ to $\text{B}$.

An excursion involves renting the boat and the bus. The cost to rent the boat is $\$ 200$ per hour, and the cost to rent the bus is $\$ 150$ per hour.

The following table shows the average body weight, $x$, and the average weight of the brain, $y$, of seven species of mammal. Both measured in kilograms (kg).

The average body weight of grey wolves is 36 kg.

In fact, the average weight of the brain of grey wolves is 0.120 kg.

The average body weight of mice is 0.023 kg.

The marks obtained by nine Mathematical Studies SL students in their projects (x) and their final IB examination scores (y) were recorded. These data were used to determine whether the project mark is a good predictor of the examination score. The results are shown in the table.

The equation of the regression line y on x is y = mx + c.

A tenth student, Jerome, obtained a project mark of 17.

A pan, in which to cook a pizza, is in the shape of a cylinder. The pan has a diameter of 35 cm and a height of 0.5 cm.

A chef had enough pizza dough to exactly fill the pan. The dough was in the shape of a sphere.

The pizza was cooked in a hot oven. Once taken out of the oven, the pizza was placed in a dining room.

The temperature, $P$, of the pizza, in degrees Celsius, °C, can be modelled by

$$P(t)=a(2.06{)}^{-<!--\; -\; -->t}+19,t⩾<!--\; ⩾\; -->0$$

where $a$ is a constant and $t$ is the time, in minutes, since the pizza was taken out of the oven.

When the pizza was taken out of the oven its temperature was 230 °C.

The pizza can be eaten once its temperature drops to 45 °C.

Abdallah owns a plot of land, near the river Nile, in the form of a quadrilateral ABCD.

The lengths of the sides are $\text{AB}=\text{40 m, BC}=\text{115 m, CD}=\text{60 m, AD}=\text{84 m}$ and angle $\mathrm{B}\stackrel{^<!--\; ^\; -->}{\mathrm{A}}\mathrm{D}={90}^{\circ <!--\; \circ \; -->}$.

This information is shown on the diagram.

The formula that the ancient Egyptians used to estimate the area of a quadrilateral ABCD is

$\text{area}=\frac{(\text{AB}+\text{CD})(\text{AD}+\text{BC})}{4}$.

Abdallah uses this formula to estimate the area of his plot of land.

Olivia’s house consists of four vertical walls and a sloping roof made from two rectangles. The height, $\text{CD}$, from the ground to the base of the roof is 4.5 m.

The base angles of the roof are $\text{A}\stackrel{\wedge <!--\; \wedge \; -->}{\text{B}}<!--\; \; -->\text{C}={27}^{\circ <!--\; \circ \; -->}$ and $\text{A}\stackrel{\wedge <!--\; \wedge \; -->}{\text{C}}<!--\; \; -->\text{B}={26}^{\circ <!--\; \circ \; -->}$.

The house is 10 m long and 5 m wide.

The length $\text{AC}$ is approximately 2.84 m.

Olivia decides to put solar panels on the roof. The solar panels are fitted to both sides of the roof.

Each panel is 1.6 m long and 0.95 m wide. All the panels must be arranged in uniform rows, with the shorter edge of each panel parallel to $\text{AB}$ or $\text{AC}$. Each panel must be at least 0.3 m from the edge of the roof and the top of the roof, $\text{AF}$.

Olivia estimates that the solar panels will cover an area of 29 m².

A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm.

The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area.

A water container is made in the shape of a cylinder with internal height $h$ cm and internal base radius $r$ cm.

The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.

The volume of the water container is $0.5{\text{m}}^3$.

The water container is designed so that the area to be coated is minimized.

One can of water-resistant material coats a surface area of $2000\text{ c}{\text{m}}^2$.

The Tower of Pisa is well known worldwide for how it leans.

Giovanni visits the Tower and wants to investigate how much it is leaning. He draws a diagram showing a non-right triangle, ABC.

On Giovanni’s diagram the length of AB is 56 m, the length of BC is 37 m, and angle ACB is 60°. AX is the perpendicular height from A to BC.

Giovanni’s tourist guidebook says that the actual horizontal displacement of the Tower, BX, is 3.9 metres.

John purchases a new bicycle for 880 US dollars (USD) and pays for it with a Canadian credit card. There is a transaction fee of 4.2 % charged to John by the credit card company to convert this purchase into Canadian dollars (CAD).

The exchange rate is 1 USD = 1.25 CAD.

John insures his bicycle with a US company. The insurance company produces the following table for the bicycle’s value during each year.

The values of the bicycle form a geometric sequence.

During the 1st year John pays 120 USD to insure his bicycle. Each year the amount he pays to insure his bicycle is reduced by 3.50 USD.

An archaeological site is to be made accessible for viewing by the public. To do this, archaeologists built two straight paths from point A to point B and from point B to point C as shown in the following diagram. The length of path AB is 185 m, the length of path BC is 250 m, and angle $\text{A}\stackrel{\wedge <!--\; \wedge \; -->}{\text{B}}<!--\; \; -->\text{C}$ is 125°.

The archaeologists plan to build two more straight paths, AD and DC. For the paths to go around the site, angle $\text{B}\stackrel{\wedge <!--\; \wedge \; -->}{\text{A}}<!--\; \; -->\text{D}$ is to be made equal to 85° and angle $\text{B}\stackrel{\wedge <!--\; \wedge \; -->}{\text{C}}<!--\; \; -->\text{D}$ is to be made equal to 70° as shown in the following diagram.