Question

A tunnel has a semicircular cross-section

and a diameter of 10 m. If the roof of a bus
just touches the roof of the tunnel when
the bus is 2 m from one side, how high is
the bus?

this question is from the IGCSE maths student book1, shape and space 2 for Edexcel ​

Answer 1

Height of bus = 4 m

Explanation:

I have attached an image showing the cross section of this tunnel.

Since the diameter is given as 10m.

Thus, radius = 10/2 = 5 m

Since radius is 5 m, it means that the distance from the center of the circle to the tunnel edge = 5 m

If the bus is 2 m from one side, it means that the centre of the road to the right hand side of the bus will be;

5 - 2 = 3 m

Looking at the image and our values here, it is clear that, the distance from the centre of the road to the point where the bus just touches the roof tunnel = 5

Using Pythagoras theorem, we can find the height of the bus, h;

h² + 3² = 5²

h² + 9 = 25

h² = 25 - 9

h² = 16

h = √16

h = 4 m