quadratic equation word problem

Question

To create some space to work under the front end of a car, a mechanic drives it up steel ramps. The ramp in the illustration is 1 foot longer than the back, and the base is 2 feet longer than the back of the ramp. Find the length of each side of the ramp.

ft (smallest value)

ft

ft (largest value)

Answer 1

From the way your problem is worded it sounds like

the inclined ramp is shorter than the base... however,

since the hypotenuse is the longest side of a right

triangle, that is not going to be correct.

 

This problem requires the two sides of the right

triangle formed by a ramp to be  b and b+1 with

the hypotenuse (inclined ramp) having length b+2

 

Using Pythagorean Theorem a²+b²=c²

where a and b are the sides of a right triangle

and c is the hypotenuse we have:

 

b² + (b+1)² = (b+2)²  with b being the back of the ramp.

 

b² + b² + 2b + 1 = b² + 4b + 4

2b² + 2b + 1 = b² + 4b + 4

b² -2b -3 = 0

 

We now have a quadratic equation which is factorable.

The factors of -3 that add to -2 are (-3)(1) so we have:

 

(b-3)(b+1) = 0

 

Either: b-3 = 0 and b = 3 ft

 

Or: b+1=0 and b = -1 ft

 

Since the length of a side will not be negative,

the length of the back of the ramp is 3 ft

 

The base or bottom, b+1, is 4 ft

 

The inclined ramp, b+2, is 5 ft.