The angle of elevation of a ladder leaning against a wall is 45°. If the distance from the base of the ladder to the wall is 15 feet, find the length of the ladder.
You have a 45° right triangle, so the legs are of equal length.
The ladder forms the diagonal of a 15' square, so the length is 15√2.
To find the length of the ladder, we can use the trigonometric concept of sine.
Let's label the length of the ladder as 'L'.
We know that the angle of elevation is 45°, and the distance from the base of the ladder to the wall is 15 feet.
The trigonometric function that relates the angle of elevation and the side opposite the angle (in this case, the length of the ladder) is sine.
The sine of an angle is defined as the ratio of the side opposite the angle to the hypotenuse.
In this case, the hypotenuse is the length of the ladder, and the side opposite the angle is the distance from the base of the ladder to the wall.
So, we can write the equation as:
sin(45°) = opp/hyp
sin(45°) = 15/L
Now, we can solve for L by isolating it:
L * sin(45°) = 15
L = 15 / sin(45°)
Using a calculator, we can find that sin(45°) is approximately 0.707.
L = 15 / 0.707
L ≈ 21.21 feet
Therefore, the length of the ladder is approximately 21.21 feet.
To find the length of the ladder, we can use trigonometry. We have the angle of elevation (45°) and the distance from the base of the ladder to the wall (15 feet).
Let's consider the triangle formed by the ladder, the wall, and the ground. The angle of elevation of 45° is the angle between the ground and the ladder. The distance from the base of the ladder to the wall is the length of the adjacent side of the triangle. Let's call it 'x'. The length of the ladder is the length of the hypotenuse of the triangle. Let's call it 'l'.
Using trigonometry, we can set up the equation:
cos(45°) = x / l
Since the cosine of 45° is 1/√2, we can rewrite the equation as:
1/√2 = x / l
To solve for 'l', we can cross-multiply:
√2 * x = l
Now, we substitute the given value of 'x' (15 feet):
√2 * 15 = l
Calculating this expression, we find:
l ≈ 21.21 feet
Therefore, the length of the ladder is approximately 21.21 feet.