To calculate the length $r$ in the cobweb diagram, we can use trigonometry. The right triangle in the diagram has an angle of $28^\circ$ and an opposite side of 41 mm.

Since we need to find the hypotenuse $r$, we use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ where:

• $\theta = 28^\circ$
• opposite = 41 mm
• hypotenuse = $r$

Rearranging for $r$: $r = \frac{\text{opposite}}{\sin(\theta)} = \frac{41}{\sin(28^\circ)}$

Calculating this: $r \approx \frac{41}{0.4695} \approx 87.3 \text{ mm}$

So, the length of $r$, rounded to 1 decimal place, is approximately 87.3 mm.

Would you like further details on the solution, or have any questions?

Here are some related questions for deeper understanding:

1. How would you calculate $r$ if the angle were changed to $45^\circ$?
2. What if the length of the opposite side was different, say 50 mm, while keeping the angle $28^\circ$?
3. How would we approach this if we were only given the adjacent side length instead of the opposite side?
4. How could you find the angle if $r$ and the opposite side were both given?
5. What other trigonometric ratios could be used if we knew two sides instead of an angle?

Tip: Remember, the sine function is especially useful in right triangles when you know an angle and the length of the side opposite to it.