help with word problem

Question

A 13-foot ladder is placed so that it reaches to a point on the wall that is 2 feet higher than twice the distance from the base of the wall to the base of the ladder.

 

a. How far from the wall is the base of the ladder?
The base of the ladder is _________ feet from the wall

 

b. How high does the ladder reach?
The ladder reaches ______ feet high

 

 

Answer 1

Hi Britney.

 

This one is a little tricky. What you're essentially looking at is a right triangle with a hypotenuse of 13. You don't know the lengths of either side, though.

 

Let's call a variable to help us out:

 

X = distance from the wall to the base of the ladder

 

We know that the distance up from the ground to the top of the ladder is 2 feet higher than twice the distance from the base.

 

That tells us:

 

Distance from the ground up = 2*X + 2

 

So what you have now is a right triangle with one side length of X, one side length of (2X+2), and a hypotenuse of 13.

 

You can use the Pythagorean Theorem to solve for X. You know that side 1 squared plus side 2 squared equals hypotenuse squared.

 

(X)^2 + (2X+2)^2 = (13)^2

 

X^2 + 4X^2 + 8X + 4 = 169

 

5X^2 + 8X - 165 = 0

 

This is a tricky looking quadratic, so it's best to use the quadratic formula to solve it.

 

a = 5

b = 8

c = -165

 

X = (-8 + √(8² - 4*5*(-165))) / (2*5)

 

X = (-8 + √(64 + 3300))/10

 

X = (-8 + 58)/10

 

X = 50/10

 

X = 5

 

I'm not going to bother with the minus root, since -8 minus another number will give us a negative value for length, which is impossible.

 

You've now determined that 5 is the distance from the wall to the base of the ladder.

 

From our equation earlier, the distance from the floor to the top of the ladder is thus:

 

2(5) + 2 = 12

 

a) 5 feet

b) 12 feet

 

Hope this helps.