11. An underground parking lot is being constructed 8.00 m below ground level. a) If the exit ramp is to rise at an angle of $$ 15^{\circ} $$, how long will the ramp be? Round your answer to the nearest hundredth of a metre b) What horizontal distance, to the nearest hundredth of a metre, is needed for the ramp?

Question

UpStudy AI · Accepted Answer

To solve the problem, we can use some basic trigonometry.

### Part a: Length of the Ramp

We know that the ramp rises at an angle of $$ 15^{\circ} $$ and the vertical height it needs to cover is $$ 8.00 \, \text{m} $$. We can use the sine function, which relates the opposite side (the height) to the hypotenuse (the length of the ramp).

The formula is:

$$
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
$$

Where:
- $$\theta = 15^{\circ}$$
- $$\text{opposite} = 8.00 \, \text{m}$$
- $$\text{hypotenuse} = L$$ (length of the ramp)

Rearranging the formula gives us:

$$
L = \frac{\text{opposite}}{\sin(\theta)} = \frac{8.00 \, \text{m}}{\sin(15^{\circ})}
$$

Calculating $$\sin(15^{\circ})$$:

$$
\sin(15^{\circ}) \approx 0.2588
$$

Now substituting this value into the equation:

$$
L = \frac{8.00 \, \text{m}}{0.2588} \approx 30.90 \, \text{m}
$$

So, the length of the ramp is approximately **30.90 m**.

### Part b: Horizontal Distance Needed for the Ramp

To find the horizontal distance, we can use the cosine function, which relates the adjacent side (the horizontal distance) to the hypotenuse (the length of the ramp).

The formula is:

$$
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
$$

Where:
- $$\theta = 15^{\circ}$$
- $$\text{adjacent} = d$$ (horizontal distance)
- $$\text{hypotenuse} = L$$ (length of the ramp)

Rearranging the formula gives us:

$$
d = L \cdot \cos(\theta) = 30.90 \, \text{m} \cdot \cos(15^{\circ})
$$

Calculating $$\cos(15^{\circ})$$:

$$
\cos(15^{\circ}) \approx 0.9659
$$

Now substituting this value into the equation:

$$
d = 30.90 \, \text{m} \cdot 0.9659 \approx 29.85 \, \text{m}
$$

So, the horizontal distance needed for the ramp is approximately **29.85 m**.

### Summary of Answers:
a) Length of the ramp: **30.90 m**  
b) Horizontal distance needed: **29.85 m**
a) Length of the ramp: 30.90 m b) Horizontal distance needed: 29.85 m

An underground parking lot is being constructed 8.00 m below
ground level.
a) If the exit ramp is to rise at an angle of , how long will the
ramp be? Round your answer to the nearest hundredth of a metre
b) What horizontal distance, to the nearest hundredth of a metre, is
needed for the ramp?

Upstudy AI Solution

Answer

Solution

Part a: Length of the Ramp

Part b: Horizontal Distance Needed for the Ramp

Summary of Answers:

Beyond the Answer

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An underground parking lot is being constructed 8.00 m below ground level. a) If the exit ramp is to rise at an angle of , how long will the ramp be? Round your answer to the nearest hundredth of a metre b) What horizontal distance, to the nearest hundredth of a metre, is needed for the ramp?