1. ## Help with ladder triangle problem

A 12-foot ladder is leaning across a fence and is touching a higher wall located 3 feet behind the fence. The ladder makes an angle of 60 degrees with the ground. Find the distance from the base of the ladder to the bottom of the fence.

I am having difficulties with this problem. I think it is 9. Because the entire ladder subtracted from 3 = 9 feet.

Are my calculations right?

2. Originally Posted by KevinVM20
A 12-foot ladder is leaning across a fence and is touching a higher wall located 3 feet behind the fence. The ladder makes an angle of 60 degrees with the ground. Find the distance from the base of the ladder to the bottom of the fence.

I am having difficulties with this problem. I think it is 9. Because the entire ladder subtracted from 3 = 9 feet.

Are my calculations right?
If the ladder were flat (on the ground) you would be correct,
HOWEVER,
"The ladder makes an angle of 60 degrees with the ground."

Make a sketch (it does not have to be to scale, nor on paper, just create a mental image) of what you have & data given.

The ladder, wall, & ground represent the three sides of a triangle.
You know the length of the hypotenuse, that is the length of the ladder or 12 feet.

The length of the base is:
$\displaystyle = 12 \times \cos (30deg)$

There are a few sin/cos values you should memorize.
sin(30) & cos(60) is one of them.

Subtract the distance the fence is from the building, from the base of the triangle, the distance the foot of the ladder is from the building.

3. 12 x Cos(30 degrees)

Cos 30 = 0.86602540378444 x 12 Length of ladder = 10.395048

that does not solve this problem..

4. Originally Posted by aidan
...

The length of the base is:
$\displaystyle = 12 \times \cos (\bold{\color{red}60^\circ})$
...

Originally Posted by KevinVM20
12 x Cos(30 degrees)

Cos 30 = 0.86602540378444 x 12 Length of ladder = 10.395048

that does not solve this problem..
1. Did you make a sketch as aidan suggested?

2. If so you must have noticed that aidan made a tiny mistake (see correction)

3. According to aidan's suggestion you've learned that $\displaystyle \cos(60^\circ)=\dfrac12$

4. Now you now that

$\displaystyle x = 12 \cdot \cos(60^\circ) = 6$

5. Therefore the foot of the ladder has a distance of 3' to the bottom of the fence.