To solve either problem making a picture is a good first step.
H |(Top of the 10 foot ladder)
O |
U |
S |
E |__________________(Base of the 10 foot ladder)
-------------------Angle here
A - We are told the height from the point to the ground is 9.2 feet and we have a 10 foot ladder. The ladder is placed from ground to point, at an angle. (That angle is what we seek).
Here we use right triangle trigonometry or SOHCAHTOA to figure out the angle. The hypotenuse is the long side and that's our 10 foot ladder. The house's height is 9.2 feet and is opposite the angle (it makes contact with the ground and the ladder). Opposite and Hypotenuse go to S, for sine. Let θ be the measure of the angle.
sin θ = 9.2 /10
sin θ = .92
sin⁻¹ (sin θ) = sin⁻¹ (.92) Take the inverse sine, sin⁻¹, of both sides
θ = sin⁻¹ (.92)
Using a calculator's inverse sine key, we have that θ equals 66.926 degrees. Using the question's rounding, we have a 67 degree angle.
B-- Now we told by OSHA that you need a 75 degree angle down below. The angle is 75° but we instead seek the proper height. Again we use SOHCAHTOA and make a picture.
H |(Top of the 10 foot ladder)
O |
U |
S |
E |__________________(Base of the 10 foot ladder)
-------------------75° Angle
We seek the side of the house - call that D (for distance). Again, we use the sine function since we know the opposite angle and the hypotenuse.
Sin 75° = D / 10 Where we put in for the angle, D, and 10.
10 sin 75° = D By multiplying both sides by 10
10 * 0.96592 = D By evaluating the sine of 75°
9.65925 = D
To the nearest tenth, we must put the top of the ladder 9.7 feet above the ground.
