A farmer uses a conveyor belt to move grain from ground level into a storage silo. The conveyor has a length of 9 metres. Its angle of elevation can be adjusted from 5° to 20°. The silo has an opening that can accommodate the conveyor belt at its lowest and highest settings. Determine the size of the opening of the silo, to the nearest tenth of a metre.

Assuming zero thickness for the belt, grain and accompanying hardware, the height of the top of the opening is

hi = 9*sin20° = 3.078m

So, the height at 5° is lo = 9*sin5° = 0.784m

hi-lo = 2.294m

To determine the size of the opening of the silo, we need to find the vertical distance (height) covered by the conveyor belt as its angle of elevation changes.

First, let's find the vertical distance covered by the conveyor belt when its angle of elevation is 5°.

We can use the trigonometric function tangent (tan) to find this distance:

tan(5°) = vertical distance / horizontal distance
tan(5°) = vertical distance / 9m

Rearranging the formula, we get:

vertical distance = tan(5°) × 9m

Using a scientific calculator, we find:

vertical distance ≈ 0.087 × 9m ≈ 0.783m

Therefore, when the angle of elevation is 5°, the conveyor belt covers a vertical distance of approximately 0.783 meters.

Next, let's find the vertical distance covered by the conveyor belt when its angle of elevation is 20°.

Following the same process as above, we have:

vertical distance = tan(20°) × 9m

Using a scientific calculator, we find:

vertical distance ≈ 0.364 × 9m ≈ 3.276m

Therefore, when the angle of elevation is 20°, the conveyor belt covers a vertical distance of approximately 3.276 meters.

To find the size of the opening of the silo, we need to consider the maximum vertical distance covered by the conveyor belt, which is 3.276 meters.

Therefore, the size of the opening of the silo should be at least 3.276 meters to accommodate the conveyor belt at its highest setting.

To determine the size of the opening of the silo, we need to find the height difference between the lowest and highest settings of the conveyor belt. This height difference will be the size of the opening.

Let's consider the lowest setting of the conveyor belt first. At this setting, the angle of elevation is 5°. We can use trigonometry to find the height difference between the ground and the conveyor belt.

Using the concept of opposite and adjacent sides in a right triangle, we can say that:

tan(5°) = height difference / length of the conveyor belt

Rearranging the equation, we get:

height difference = tan(5°) * length of the conveyor belt

Calculating this value:

height difference = tan(5°) * 9 m

Now, calculate the height difference using a calculator:

height difference ≈ 0.087 * 9 m
height difference ≈ 0.783 m

So, at the lowest setting of the conveyor belt, the height difference between the ground and the conveyor belt is approximately 0.783 m.

Now, let's consider the highest setting of the conveyor belt. At this setting, the angle of elevation is 20°. Again, we can use trigonometry to find the height difference.

Using the same approach as before:

height difference = tan(20°) * length of the conveyor belt

Calculating this value:

height difference = tan(20°) * 9 m

Now, calculate the height difference using a calculator:

height difference ≈ 0.363 * 9 m
height difference ≈ 3.267 m

So, at the highest setting of the conveyor belt, the height difference between the ground and the conveyor belt is approximately 3.267 m.

Therefore, the size of the opening of the silo is the difference between the highest and lowest height differences, which is approximately:

3.267 m - 0.783 m ≈ 2.484 m

Thus, the size of the opening of the silo is approximately 2.5 meters to the nearest tenth of a meter.