Conveyor belts called grain elevators are used to move grain into a silo. Answer the following questions knowing that the lower end of the belt is 100 feet from the base of the silo and that the silo is 150 feet tall.

a. How long is the belt?
b. If we know the angle of elevation from the lower end of the belt to a window on the side of the silo is 45°, use special right triangle ratios to calculate the height of the window from the ground.
c. We need a ramp to the window. How long would it need to be?

draw some diagrams first to help visualize (if you need to)

a.
so one leg is 100ft and the other is 150ft
use Pythagorean theorem
for legs a and b and hytponuse length c
a²+b²=c²
so
given legs 100 and 150
100²+150²=c²
10000+22500=c²
32500=c²
sqrt both sides
50√13=c
the belt is 50√13 ft

b.
so, we take the point of the lower end of the belt and connect it to the window with a line
the base of the belt is 100ft ffrom the silo
since it is 45 degerees, and the base is 100ft, the height is 100ft as well
the height of the window from the ground is 100ft

c. ok, use special right triangles
45-45-90 triangle
if one leg is x then the hypontuse is x√2
so one leg is 100ft so the hytponuse is 100√2 ft
the ramp must be 100√2 ft

a. 50√13 ft
b. 100 ft
c. 100√2 ft

oeerivona oeerivona · Answer 2 · 2021-08-22T12:55:36+02:00

A special right triangle is one that has features that simplify or make it easier to calculate its dimensions

a. The length of the belt is approximately 180.28 feet

b. The height of the window is 100 feet

c. The length of the ramp is approximately 141.42 feet

The reason for the above selection is as follows:

The given parameters are;

The (horizontal) distance of the belt from the base of the silo, d = 100 feet

The height of the silo, h = 150 feet

a. Required; The length of the belt

The length of the belt, the height of the silo, h, and the horizontal distance, d, of the end of the belt from the base of the silo form a right triangle

The legs of the right triangle = The vertical height and the horizontal distance

The hypotenuse side = The length of the belt, l

According to Pythagoras's theorem, we have;

The hypotenuse side, I = √(h² + d²)

∴ l = √(100² + 150²) = 50·√(13) ≈ 180.28

The length of the belt, l ≈ 180.28 feet

b. The given parameter;

The angle of elevation from the belt lower end to a window on the silo, θ = 45°

Required: Calculate the height of the window from the ground with special triangles relationships

Solution:

One of the angles of the right triangle formed by the line of the angle of elevation to the window, θ₁ = 45°

The other angle of the right triangle, θ₂ = 180° - 90° - 45° = 45°

θ₁ = 45° = θ₂, therefore;

The special relationship = The triangle formed is an isosceles right triangle, and the legs of the triangle are equal

The legs of the triangle are;

1) The height of the window, from the ground

2) The horizontal distance of the lower of the end of the belt to the base of the silo

∴ The height of the window, [tex]h_w[/tex]= Horizontal distance of belt, d = 100 feet

The height of the window, [tex]h_w[/tex] = 100 feet

c. Required: The length of the ramp to the window

Let R represent the length of the ramp, we have;

According to Pythagoras's theorem, we have;

R² = [tex]\mathbf{h_w}[/tex]² + d²

lugging in the vales, gives;

R² = 100² + 100² = 200,000

∴ R = 100·√2

The length of the ramp, R = 100·√2 feet ≈ 141.42 feet

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