Help with ladder triangle problem

KevinVM20 · #1 $Old$ July 4th, 2009, 04:01 AM

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?

aidan · #2 $Old$ July 4th, 2009, 07:40 AM

Quote:

Originally Posted by KevinVM20 $View Post$

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:
$= 12 \times \cos (30deg)$

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.

KevinVM20 · #3 $Old$ July 5th, 2009, 11:54 PM

12 x Cos(30 degrees)

Cos 30 = 0.86602540378444 x 12 Length of ladder = 10.395048

that does not solve this problem..

earboth · #4 $Old$ July 6th, 2009, 12:13 AM

Quote:

Originally Posted by aidan $View Post$

...

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

...

Quote:

Originally Posted by KevinVM20 $View Post$

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 $\cos(60^\circ)=\dfrac12$

4. Now you now that

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

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