Question

A 34-foot bike ramp needs to be set up such that the horizontal length along the ramp is four times that of the height of the ramp. Write and solve a system to find the horizontal length and height for the ramp. If necessary, round answers to the nearest hundredth.

a) Set up the system of equations representing the height and horizontal length.
b) Solve the system of equations to find the height and horizontal length.
c) Discuss the practical implications of the solution in the context of setting up the bike ramp.
d) Compare the obtained solution to a scenario where the horizontal length is twice the height.

Answer 1

To find the height and horizontal length of the ramp, we can set up and solve a system of equations. The height of the ramp is approximately 6.8 feet and the horizontal length along the ramp is approximately 27.2 feet.

Step-by-step explanation:

To solve this problem, we can create a system of equations representing the height and horizontal length of the bike ramp:

Let h be the height of the ramp

Let x be the horizontal length along the ramp

From the problem, we know that the horizontal length along the ramp is four times that of the height of the ramp. This gives us the equation:

x = 4h

We also know that the total length of the ramp is 34 feet. This gives us the equation:

h + x = 34

To solve this system of equations, we can substitute the value of x from the first equation into the second equation:

h + 4h = 34

Combining like terms, we get:

5h = 34

Dividing both sides by 5, we find that:

h = 6.8

Substituting this value of h back into the first equation, we get:

x = 4 * 6.8 = 27.2

Therefore, the height of the ramp is approximately 6.8 feet and the horizontal length along the ramp is approximately 27.2 feet.