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It looks like you've uploaded an image containing several math problems. I'll extract and address each question so that I can help you with any of them. Here are the problems from the image:

1. **If \(\frac{1}{a} + \frac{1}{b} = \frac{1}{c}\), then prove that \(a + b + c = abc\) when each ratio equals \(\frac{1}{d}\).**
   
2. **Two rods A and B are 10 feet and 8 feet long, respectively. Both rods are bent into circles. If 7 such circles of rod A and 8 such circles of rod B have the same total area, find the ratio of their diameters.**

3. **A clock shows 3:15. What is the angle between the hour and minute hands?**

4. **Given that \(x^3 + y^3 = z^3\) and \(x + y + z = 0\), prove that \(x^3 = 3xyz\).**

5. **If \(x + \frac{1}{x} = 2\), find the general form of \(x^n + \frac{1}{x^n}\).**

6. **Show that \(x^2 + y^2 + z^2 - xy - yz - zx = \frac{1}{2}((x-y)^2 + (y-z)^2 + (z-x)^2)\).**

7. **Let \( \vec{a} \) be a unit vector and \( \vec{b} \) be any vector. Show that \((\vec{a} \times \vec{b}) \cdot \vec{b} = 0\).**

8. **Find the inverse of the matrix \(A = \begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix}\).**

9. **A straight wire of length 18m is bent into a circular arc subtending an angle of \(60^\circ\) at the center of the circle. Calculate the radius of the circle.**

10. **Find the derivative of \(y = x^x\) with respect to \(x\).**

11. **Determine \( \frac{d}{dx} \) of \( e^{\tan^{-1}(x)} \).**

12. **Solve for \(x\) in the equation \(x^2 - 6x + 5 = 0\).**

13. **Explain why the sum of the first 1000 natural numbers is 500500.**

14. **If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are \(x_1\) and \(x_2\), prove that \(x_1 + x_2 = -\frac{b}{a}\) and \(x_1x_2 = \frac{c}{a}\).**

15. **A particle moves along the x-axis according to the law \(x(t) = t^3 - 6t^2 + 9t + 5\). Determine when the particle is at rest.**

Please let me know which problem you'd like to start with, or if you want solutions or explanations for multiple questions.

If \(\frac{1}{a} + \frac{1}{b} = \frac{1}{c}\), then prove that \(a + b + c = abc\) when each ratio equals \(\frac{1}{d}\).

Learn how to prove the equation a + b + c = abc given that \(\frac{1}{a} + \frac{1}{b} = \frac{1}{c}\) using rational equation concepts. This detailed explanation involves algebraic manipulation and is suitable for high school students in Grades 10-12.

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