In the figure, Katie walks 32 m down path AB with gradient 1 : 5 and then walks 45 m down path BC with gradient 1: 3. Find the horizontal distance between A and C, correct to 3 significant figures

To solve this problem, we need to break it down into two parts, corresponding to the two paths Katie walks, AB and BC. Since we know the gradients and the distances, we can use trigonometry to find the horizontal distances for each path and then sum them to find the total horizontal distance between points A and C.

### Path AB
1. **Gradient of AB**: 1:5. This means for every 1 unit of vertical distance, there are 5 units of horizontal distance.
   
2. **Distance of AB**: 32 m.

Using the Pythagorean theorem for a right-angled triangle, the ratio given by the gradient is horizontal:vertical = 5:1.

Let's denote:
- Horizontal distance as \( x \).
- Vertical distance as \( y \).

The gradient ratio tells us:
\[
\frac{y}{x} = \frac{1}{5} \implies y = \frac{x}{5}
\]

Using the distance formula:
\[
x^2 + y^2 = 32^2
\]
\[
x^2 + \left(\frac{x}{5}\right)^2 = 1024
\]
\[
x^2 + \frac{x^2}{25} = 1024
\]
\[
\frac{26x^2}{25} = 1024
\]
\[
26x^2 = 25600
\]
\[
x^2 = \frac{25600}{26}
\]
\[
x^2 = 984.615
\]
\[
x \approx 31.4 \text{ m}
\]

### Path BC
1. **Gradient of BC**: 1:3. This means for every 1 unit of vertical distance, there are 3 units of horizontal distance.

2. **Distance of BC**: 45 m.

Using the gradient ratio horizontal:vertical = 3:1, we have:
\[
\frac{y}{x} = \frac{1}{3} \implies y = \frac{x}{3}
\]

Using the distance formula:
\[
x^2 + y^2 = 45^2
\]
\[
x^2 + \left(\frac{x}{3}\right)^2 = 2025
\]
\[
x^2 + \frac{x^2}{9} = 2025
\]
\[
\frac{10x^2}{9} = 2025
\]
\[
10x^2 = 18225
\]
\[
x^2 = 1822.5
\]
\[
x \approx 42.7 \text{ m}
\]

### Total Horizontal Distance from A to C
\[
x_{\text{total}} = x_{AB} + x_{BC}
\]
\[
x_{\text{total}} = 31.4 + 42.7
\]
\[
x_{\text{total}} \approx 74.1 \text{ m}
\]

Therefore, the horizontal distance between A and C is approximately **74.1 m**, correct to 3 significant figures.

---

Would you like further details, or do you have any other questions?

Here are some related questions that you might find interesting:

1. How can we calculate the vertical distance Katie walked along each path?
2. What is the total vertical distance Katie descended from A to C?
3. How would the horizontal distance change if the gradient of path AB was 1:4 instead?
4. What would be the horizontal distance if Katie walked a third path CD with a different gradient?
5. How can gradients be used to calculate slope angles in degrees?

**Tip**: Remember, gradients express the ratio of vertical to horizontal distances, and using trigonometry with these ratios can simplify finding distances and angles.

Learn how to calculate the horizontal distance between points A and C using gradients 1:5 and 1:3, with detailed steps and formulas. Solve for the total horizontal distance accurately to 3 significant figures.

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