Hard Geometry problem help?

Question

A conveyor belt is to be set up to move bundles of shingles up to a roof. If the edge of the roof is 10 feet above the ground and the conveyor can make no more than a 45 degree angle with the ground, how long a conveyor belt will be needed? How far away from the house will the bottom of the conveyor be placed?
What formula am I suppose to use?

Answer 1

Draw a triangle with one corner at the edge of the roof, one on the ground directly below that, and one on the ground where the base of the conveyor is placed. This will give you a right-angled triangle with the conveyor as the hypotenuse.

The angle between the conveyor and the ground is 45°. The side length opposite this angle is 10 feet. The conveyor is the hypotenuse, so to get the length of the conveyor we use
sin 45° = opposite / hypotenuse = 10 / x
so x = 10 / sin 45° = 10√2 = 14.1 ft (1 d.p.)

The distance from the house to the base of the conveyor is the adjacent side, so we use
tan 45° = opposite / adjacent = 10 / y
so y = 10 / tan 45° = 10 ft.

Answer 2

This is a problem in basic trigomentry. If the conveyor can make no more that a 45° angle with the ground, then the length of the conveyor would max out at that angle. Also, the height above ground and the distance from the house will be the same, 10 ft.

Use the Pythagorean Theorem to solve:
x = 10
y = 10
length, or L = √(x² +y²)
L = √(10 ² + 10²
L = √(100 + 100)
L = √200
L ≈ 14.14 ft.

Answer 3

If the angle is shorter, the belt has to be longer. If we want the smallest length of belt, we have to use the maximum angle of 45 degrees.

Notice that the building forms a right angle to the ground, and with the belt leaning up against the edge of the roof, this gives you a 45-45-90 triangle where the hypotenuse is the belt. This means the belt length is √2 times one of the legs, so the belt has to be 10√2 feet long.

Answer 4

Draw a triangle.
Height is 10 feet (height of edge of roof).
Angle at the ground (between ground and belt) is 45 deg.

Presumably there's a right angle between the house and the ground, so the angle between the house and the belt (at the roof) is 45 deg.

That makes it an isosceles triangle, so the bottom edge (along the ground) is also 10 feet. The belt is the hypotenuse of the right angled triangle (so Pythagoras gives you its length).