Lombard St. in San Francisco, known for its famous curves, has a 27% grade. This means that
the slope is −
27
100
. In Tennessee, route 297 through Big South Fork National Recreation Area has a 13% grade.
Driving on this road you notice elevation signs that say you have descended 1000 feet. What is your change in
your horizontal distance in miles?

To calculate the change in horizontal distance when you descend 1000 feet on a road with a 13% grade, you can follow these steps:

### Step 1: Understand the grade
- A 13% grade means that for every 100 feet of horizontal distance, the elevation changes by 13 feet.
- The slope is \( \text{slope} = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{13}{100} \).

### Step 2: Calculate the horizontal distance
- You're given that the vertical descent is 1000 feet.
- Let \( x \) be the horizontal distance in feet.

Using the slope formula:
\[
\frac{13}{100} = \frac{1000}{x}
\]

### Step 3: Solve for \( x \)
- Cross-multiply to solve for \( x \):
\[
13x = 100 \times 1000
\]
\[
13x = 100000
\]
\[
x = \frac{100000}{13} \approx 7692.31 \text{ feet}
\]

### Step 4: Convert feet to miles
- There are 5280 feet in a mile, so convert the horizontal distance to miles:
\[
\text{Horizontal distance in miles} = \frac{7692.31}{5280} \approx 1.46 \text{ miles}
\]

### Final Answer
The change in your horizontal distance is approximately **1.46 miles**.

Would you like further details or have any questions?

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Here are 8 related questions that expand on this information:

1. What is the change in horizontal distance if the descent was 2000 feet on the same grade?
2. How would the horizontal distance change if the grade was increased to 20%?
3. Can you calculate the horizontal distance change for a descent of 500 feet on a 15% grade?
4. What would be the effect on the horizontal distance if the road had a lower grade, say 10%?
5. How does the slope formula apply to calculating road distances in other scenarios?
6. What is the physical significance of a road grade, and how does it impact driving?
7. How do you convert between different units of distance, such as feet and miles?
8. What tools can drivers use to estimate elevation changes on roads?

**Tip:** Understanding road grades is essential for safe driving, especially in mountainous or hilly areas where steep grades can significantly affect vehicle control.

Explore how to calculate the change in horizontal distance when descending on roads with varying grades. Using examples from Lombard St. in San Francisco and route 297 in Tennessee, learn how to apply the slope formula and proportions to determine horizontal distance changes. Understand the impact of road grades on driving and safety. Suitable for high school students and those interested in practical applications of mathematics in road engineering.

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