Let's solve the problem step by step.

Problem Recap:

We have a bridge consisting of two straight sections, each hinged at one end. When both sections are raised 6 meters above the horizontal, each section has a gradient (slope) of 0.8. We need to find the distance $d$, which is the distance between the closest end points of the two sections.

Step 1: Understanding the Gradient

The gradient (or slope) is given as 0.8. This means: $\text{Gradient} = \frac{\text{Vertical Rise}}{\text{Horizontal Run}} = 0.8$ Let the horizontal run be $x$. Then: $\frac{6 \text{ m}}{x} = 0.8$ So, $x = \frac{6}{0.8} = 7.5 \text{ m}$

Step 2: Determining the Distance $d$

Each section of the bridge forms a right triangle with the vertical rise of 6 meters and the horizontal run of 7.5 meters.

The distance $d$ between the closest end points of the two sections is simply the horizontal distance between the tops of the two sections, which is $2x$ since there are two identical sections, one on each side.

Thus: $d = 2 \times 7.5 = 15 \text{ meters}$

Conclusion:

The distance $d$ between the closest end points of the two sections is 15 meters.

Would you like further details or have any questions?

Here are some related questions:

1. How would the problem change if the gradient was different?
2. What would be the effect on $d$ if the vertical rise was greater than 6 meters?
3. Can you find the length of each section of the bridge?
4. How would the solution change if the sections of the bridge were not equal in length?
5. What if the bridge had more than two sections with different slopes?

Tip: When working with gradients, remember that they describe the ratio of vertical rise to horizontal run, which is key in solving problems involving slopes or angles.