Solve a practical right-angled-triangle problem

A straight loading rail rises from A to B. F is directly below B and level with A. Which physical length is the hypotenuse? The sketch is not to scale

A straight 8 m rail is raised from 30° to 60° above horizontal. Predict what changes

Increase the angle from its starting position, then decrease it. Watch the rise and run while the sloping length stays fixed at 8 m

As the angle increases above the starting position, what happens to the two components?

Make the physical quantities clear

Use the right-triangle methods you already know. The new decision is which physical measurement belongs to each side and which angle the problem gives

1. 1Define the endpoints Decide what each side measures. A height above the observer is different from a height above the ground
2. 2Use the marked angle Opposite and adjacent depend on the stated angle. The hypotenuse is always opposite the right angle. An angle from the vertical can be used if the side roles match it
3. 3Choose from the known and wanted sides Use the ratios from the earlier topics. Keep all lengths in a common unit \(\displaystyle \begin{aligned}\sin\theta&=\frac{o}{h}\\\cos\theta&=\frac{a}{h}\\\tan\theta&=\frac{o}{a}\end{aligned}\)
4. 4Interpret the answer Report the requested physical quantity. A hypotenuse must exceed either leg. Keep the full calculator value until the final rounding step

Rail length AB

\(\displaystyle 6.4\ \mathrm{m}\)

Angle from the horizontal at A

\(\displaystyle 41^{\circ}\)

A straight guide rail runs from A to B. Calculate its vertical rise FB to one decimal place. The sketch is not to scale

Model

\(\displaystyle a=7.2\cos38^{\circ}\)

A straight 7.2 m rail makes an angle of 38° to horizontal. Find its horizontal run to one decimal place

Model

\(\displaystyle H=18\tan32^{\circ}\)

A vertical mast stands on level ground. Its top is seen at 32° above horizontal from a point on the ground 18 m from its base. Find the mast height to one decimal place

Model

\(\displaystyle H=1.4+18\tan32^{\circ}\)

The 32° angle is measured from an instrument 1.4 m above level ground, 18 m from the mast. Find the total mast height to one decimal place

Model

\(\displaystyle x=6.5\sin42^{\circ}\)

A 650 cm straight brace makes a 42° angle with a vertical wall. Find its horizontal reach in metres to two decimal places

A slope model gives a 6.4 m hypotenuse and a 4.2 m vertical rise. What is the sensible interpretation?