quadratic equations

Question

The inclined ramp of the boat launch shown is 8 meters longer than the rise of the ramp. The run is 7 meters longer than the rise. How long are the three sides of the ramp?
rise
m
run
m
ramp
m

Answer 1

If you draw this out you have a right triangle

The vertical side for the rise has length r

The run is length r + 7

The inclined ramp, forming the hypotenuse, has length r + 8

 

By the Pythagorean Theorem  a²+b²=c² where a, b are the sides

and c is the hypotenuse of a right triangle so:

 

r² + (r+7)² = (r+8)²

r² + r² + 14r + 49 = r² + 16r + 64

2r² + 14r + 49 = r² + 16r + 64

r² -2r - 15 = 0

 

We have a quadratic equation, but this can be factored.

The factors of -15 that add to -2 are -5(3) so this factors to:

 

(r-5)(r+3) = 0

Either r-5 = 0 and r=5    or

r+3 = 0 and r = -3

 

Since we will not have a side with a negative length

r = 5m for the rise

r + 7 = 12m for the run

r + 8 = 13m for the inclined ramp