A ladder leaning against a house makes an angle of 60 degrees with the ground. The foot of the ladder is 7 ft from the house. How long is the ladder?
14
To solve this problem, we can use the sine function. The sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse.
Let's label the length of the ladder as "L". The opposite side is the height of the ladder against the house, which is 7 ft. The hypotenuse is the length of the ladder.
Using the sine function:
sin(60 degrees) = opposite/hypotenuse
sin(60 degrees) = 7 ft/L
To solve for L, we can rearrange the equation:
L = 7 ft / sin(60 degrees)
Using a scientific calculator or referring to a trigonometric table, we can find that sin(60 degrees) is equal to √3/2.
L = 7 ft / (√3/2)
L = 7 ft * (2/√3)
L = (14/√3) ft
Rationalizing the denominator, we can multiply the numerator and denominator by √3:
L = (14 * √3) / (√3 * √3)
L = (14 * √3) / 3
Therefore, the length of the ladder is approximately (14 * √3) / 3 ft or about 8.08 ft.
To find the length of the ladder, we can use trigonometry. In this case, we can use the trigonometric function sine (sin) since we have the opposite (height from the ground) and adjacent (distance from the house) sides.
First, let's label the sides of the right triangle formed by the ladder, the ground, and the side of the house.
The side opposite to the angle of 60 degrees is the height of the ladder from the ground, which we'll call "h."
The side adjacent to the angle of 60 degrees is the distance between the foot of the ladder and the house, given as 7 ft, which we'll label as "a."
Lastly, the length of the ladder is the hypotenuse of the triangle, which we'll call "L."
Using the sine function (sin), we can write the equation: sin(60) = h / L.
Since the sine of 60 degrees is √3 / 2, the equation becomes: √3 / 2 = h / L.
We know the value of "a" is 7 ft, and we can find "h" using the equation h = a * sin(60).
h = 7 ft * (√3 / 2) = (7 * √3) / 2 = 3.5√3 ft.
Now, we can substitute the values of "h" and "L" into the equation from before: √3 / 2 = (3.5√3 ft) / L.
To solve for "L," we can multiply both sides of the equation by 2/L and then multiply both sides by L to isolate the variable:
√3 * L = 2 * (3.5√3 ft).
L = (2 * (3.5√3 ft)) / √3.
L = 7 ft.
Therefore, the length of the ladder is 7 ft.