3.2 Trigonometry in triangles questions for IB Maths AI

Find the gradient of the ramp.

Local building regulations state that the gradient of a wheelchair ramp must not be greater than $\frac{1}{12}$.

Show that Amara's ramp satisfies the regulations.

Find the angle that the ramp makes with the horizontal.

A second entrance to the library is $0.6$ m above the ground. Find the minimum horizontal distance needed for a ramp to this entrance that satisfies the regulations.

Find the size of angle PQR.

Calculate the area of the field, in m². Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Given that 1 hectare $={10}^4$ m², write down the area of the field in hectares.

Oksana plants wheat in the whole field. She expects a yield of $3.2\times {10}^3$ kg of wheat per hectare.

Find the expected total yield of wheat, in kg.

The mean mass of one grain of wheat is $4.5\times {10}^{-5}$ kg.

Estimate the number of grains of wheat in the expected yield. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Oksana sells all the wheat for 0.24 US dollars (USD) per kg.

Find the total income from the wheat. Give your answer correct to the nearest dollar.

Write down the vector $\overrightarrow{AE}$.

Find the length of the edge $[AE]$.

Find the angle between the edge $[AE]$ and the horizontal base.

The roof covering must be laid at a pitch of at least 30°, where the pitch of a face is the angle between that face and the horizontal.

Show that the face $ABFE$ meets this requirement.

Use a vector product to find the area of the triangular face $ADE$.

Find the total area of the four faces of the roof.

The space under the roof is divided into a triangular prism, between the vertical planes $x=3$ and $x=9$, and two pyramids, one at each end.

Show that the volume of the space under the roof is $40h$ m³.

Show that the total area of the roof, $R$ m², is given by $R(h)=18\sqrt{h^2+16}+8\sqrt{h^2+9}$.

The owners ask for 160 m³ of space under the roof.

Find the percentage increase in the area of the roof compared with the first design.

Because of heavy snowfall in Sapporo, the pitch of the faces $ABFE$ and $DCFE$ must be between 30° and 45° inclusive. Find the range of possible volumes of the space under the roof.

State one assumption Haruto has made in calculating the volume of the space under the roof.

Find the vertical height gained when a car travels a horizontal distance of $850$ m along this road.

Find the angle that the road makes with the horizontal.

Find the distance the car travels along the road.

Write down the angle, in radians, through which the minute hand turns in 25 minutes. Give your answer as an exact multiple of $\pi$.

Find the distance travelled by the tip of the minute hand in 25 minutes.

Find the angle, in radians, through which the hour hand turns in 25 minutes.

At exactly 16:00 the angle between the two hands is $\frac{2\pi }{3}$ radians. Find the distance between the tips of the two hands at this time.

Show that, $t$ minutes after 16:00, the angle between the two hands is $\frac{2\pi }{3}-\frac{11\pi t}{360}$ radians, for $0\le t\le 21$.

Find the time, after 16:00, at which the two hands first point in the same direction. Give your answer to the nearest second.

Find the area of the clock face swept out by the minute hand between 16:00 and the time found in part (f).

Find, in radians, the angle through which the string turns as the bob travels from one extreme position to the other.

Convert your answer to part (a) to degrees.

Find the straight-line distance between the two extreme positions of the bob. Give your answer in centimetres.