A ladder is leaning against a house and is forming a 60 degree angle with the ground. If the top of the ladder is 24 feet off of the ground, how long is the ladder? Leave your answer in exact form.
48 ft
16 radical 3 ft
24 radical 3 ft
12 ft
2007-07-20 15:06:34 · 7 answers · asked by confused 3 in Science & Mathematics ➔ Mathematics
Hi,
When you know the 60° side of a triangle, you divide its length by √3 to get the length of the 30° side. That means since our 60° side is 24, that the 30° side would be 24/√3. To simplify it multiply top and bottom both by √3 to rationalize the denominator.
........._
24 * √3
----------- =
√3 * √3
......_
24√3
-------- =
3
...._
8√3 <== This is the simplified 30° side.
Now that we know the 30° side, you double it to get the length of the hypotenuse.
......._........._
2*8√3 = 16√3 <== This is the length of the hypotenuse, answer b.
I hope that helps!! :-)
......_
16√3 is the answer.
2007-07-20 15:09:51 · answer #1 · answered by Pi R Squared 7 · 0⤊ 0⤋
We assume that the wall of the house is vertical and that the ground is horizontal. Therefore, the wall makes an angle of 90 degrees with the ground.
The ladder, leaning against the house, forms a right-angle triangle with the wall and the ground. If the angle between the ladder and the ground is 60 deg. then the angle at the wall must be 30 (sum of angles in a triangle is 180, and we had already identified a 90 degree angle between the wall and the ground).
We know (from tables or from a calculator) that the sine of 60 degrees is equal to half of SQRT(3). We also know from definitions that the sine of an angle is equal to the ratio of the opposite side over the hypothenuse.
The opposite side is 24 feet. The hypothenuse is the length of the ladder (let's call it L)
24/L = Sin(60) = (1/2) * radical 3
24/L = (1/2) * radical 3
multiply both sides by 2:
48/L = radical 3
multiply both sides by radical 3:
(48 * radical 3) / L = 3
Multiply both sides by L:
48 radical 3 = 3 L
Divide both sides by 3:
16 radical 3 = L (length of ladder)
2007-07-20 15:18:51 · answer #2 · answered by Raymond 7 · 1⤊ 0⤋
The ladder is 48 ft tall. Let me explain why. The side of the house is 24 feet tall and this is a 30 60 90 right triangle, therefore that is the x side. The hypotenuse of the triangle (the ladder) is the 2x side which makes it 48 ft.
2007-07-20 15:12:15 · answer #3 · answered by thehumm09 2 · 1⤊ 1⤋
Sides of a right angle triangle are proportional to opposite angles.
Therefore distance from base to wall is 12 feet.
Pythagorus says square on hypotenuse = sum of squares on other 2 sides.
Let "L" represent length of ladder.
L^2 = 12^2 + 24^2
L^2 = 144 + 576 = 720
L = Sqrt of 720
2007-07-20 15:20:22 · answer #4 · answered by Robert S 7 · 1⤊ 0⤋
sin60=24/x
x= 24/sin60
the ladder is 27.71feet
hope it works
2007-07-20 15:12:33 · answer #5 · answered by **PiNoY YFC** 7 · 0⤊ 0⤋
HI STEPHANIE!
2007-07-20 15:10:08 · answer #6 · answered by Anonymous · 0⤊ 3⤋
i dont know
2007-07-20 15:09:35 · answer #7 · answered by GREG S 2 · 0⤊ 3⤋