If a ladder 40 feet long is placed so as to reach a window 30 feet high,what angel does it make with the level ground,and how far is its foot from tha base of the building?
did you make a sketch?
If so you will see that
sinØ = 30/40 = .75
Ø = appr 48.6°
also:
x^2 + 30^2 = 40^2
x^2 = 700
x = 7√10 or appr 22.1 ft
To find the angle the ladder makes with the level ground and the distance of its foot from the base of the building, we can use trigonometric functions.
Let's denote the angle the ladder makes with the level ground as θ (theta) and the distance of its foot from the base of the building as x.
Using the trigonometric function sine (sin), we can establish the following relationship:
sin(θ) = opposite/hypotenuse
In this case, the opposite side is the height of the window, 30 feet, and the hypotenuse is the length of the ladder, 40 feet. Therefore, we can write:
sin(θ) = 30/40
To find the angle θ, we need to take the inverse sine (sin⁻¹) of both sides:
θ = sin⁻¹(30/40)
Using a calculator, we can find the value of θ to be approximately 45.58 degrees.
Next, to find the distance of the ladder's foot from the base of the building (x), we can use the trigonometric function cosine (cos):
cos(θ) = adjacent/hypotenuse
In this case, the adjacent side is x (the unknown we want to find) and the hypotenuse is still 40 feet. Therefore, we can write:
cos(θ) = x/40
To find x, we need to rearrange the equation:
x = cos(θ) * 40
Now, we can substitute the value of θ we found earlier:
x = cos(45.58 degrees) * 40
Using a calculator again, we can find that x is approximately 28.28 feet.
So, the ladder makes an angle of approximately 45.58 degrees with the level ground, and its foot is approximately 28.28 feet from the base of the building.