Question

Ramps help people go into buildings. A ramp that is 10 m long must not have a height greater than 0.83 m. (a) Here are the plans for a ramp: (2 mark) Is this ramp too high? You must show calculations to explain your answer. (b) Here are the plans for a different ramp How long is the base of this ramp? (2marks) You must show your calculations. (c) The recommended gradient of a ramp is 1 in 20. (2marks) What angle gives the recommended gradient? You must show your calculations. Total : 30 marks

Answer 1

(a) To determine if the first ramp is too high, we need to calculate its gradient (slope), which is given by the formula:

Gradient = Height / Length

In this case, the height should not be greater than 0.83 m, and the length of the ramp is 10 m. Let’s calculate the gradient:

Gradient = 0.83 m / 10 m = 0.083

To determine if it meets the recommended gradient, we need to calculate the reciprocal (1/Gradient):

1 / 0.083 ≈ 12.05

The calculated gradient is approximately 1 in 12.05. This is steeper than the recommended gradient of 1 in 20, so the first ramp is too high.

(b) To find the length of the base of the second ramp, we can use the Pythagorean theorem since it appears to form a right triangle:

Base² + Height² = Length²

Given the height of 0.83 m and the length of 10 m, let’s solve for the base:

Base² + (0.83 m)² = (10 m)²

Base² + 0.6889 m² = 100 m²

Base² ≈ 100 m² - 0.6889 m²

Base² ≈ 99.3111 m²

Base ≈ √99.3111 m ≈ 9.965 m

So, the length of the base of the second ramp is approximately 9.965 meters.

(c) To find the angle that gives the recommended gradient of 1 in 20, you can use trigonometry. The tangent of the angle is equal to the gradient:

tan(angle) = 1/20

To find the angle, take the arctangent (inverse tangent) of 1/20:

angle = arctan(1/20)

Using a calculator:

angle ≈ 2.86 degrees

So, the recommended gradient corresponds to an angle of approximately 2.86 degrees.