Maths biking question

Question

Brendan enjoys mountain biking. He has found that the max gradient which he can cycle up is 0.3, and the min gradient that he can safely descend is 0.5. His map has a scale of 2cm to 1km, with contours every 25m. 

 

a) What is the min distance between the contours on his map that allows him to go uphill?

 

b) What is the min distance between the contours on his map that allows him to go downhill?

Answer 1

Given:

Maximum uphill gradient = 0.3 → This means he can go up 0.3 meters vertically per 1 meter horizontally.

Minimum safe downhill gradient = 0.5 → This means he can go down 0.5 meters vertically per 1 meter horizontally.

Map scale = 2 cm represents 1 km = 1000 m.

Contour interval = 25 meters.

Part a: Uphill – Minimum Distance Between Contours

Step 1: Use gradient formula:

Gradient=vertical risehorizontal run

Gradient=horizontal runvertical rise​

We are given the gradient = 0.3 and vertical rise = 25 m (between contours).

0.3=25horizontal run

0.3=horizontal run25​

Solve for horizontal run:

horizontal run=250.3=83.33 meters

horizontal run=0.325​=83.33 meters

Step 2: Convert horizontal run to map distance:

Map scale:

2 cm:1000 m⇒21000=0.002 cm per meter

2 cm:1000 m⇒10002​=0.002 cm per meter

So,

Map distance=83.33×0.002=0.1667 cm

Map distance=83.33×0.002=0.1667 cm

Answer for a)

Minimum map distance between contours for uphill travel = 0.17 cm (rounded to 2 decimal places)

Part b: Downhill – Minimum Distance Between Contours

Step 1: Use gradient formula again:

Gradient=25horizontal run,Gradient=0.5

Gradient=horizontal run25​,Gradient=0.5

horizontal run=250.5=50 meters

horizontal run=0.525​=50 meters

Step 2: Convert to map distance:

Map distance=50×0.002=0.1 cm

Map distance=50×0.002=0.1 cm

Answer for b)

Minimum map distance between contours for downhill travel = 0.1 cm