1. **Problem (a):** Find the total surface area of a right circular cone with base diameter 20 cm and height 24 cm. - Formula for total surface area of a cone: $$\text{Total Surface Area} = \pi r^2 + \pi r l$$ where $r$ is the radius and $l$ is the slant height. - Calculate radius: $r = \frac{20}{2} = 10$ cm. - Calculate slant height using Pythagoras theorem: $$l = \sqrt{r^2 + h^2} = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26$$ cm. - Calculate total surface area: $$\pi (10)^2 + \pi (10)(26) = 100\pi + 260\pi = 360\pi$$ cm$^2$. 2. **Problem (b):** Calculate the distance of an 8 cm chord from the center of a circle with radius 5 cm. - Let the distance from center to chord be $d$. - Half chord length: $\frac{8}{2} = 4$ cm. - Use right triangle relation: $$d = \sqrt{r^2 - \left(\frac{c}{2}\right)^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$$ cm. 3. **Problem (c):** Find the distance of a 10 cm chord from the center of a circle with diameter 26 cm. - Radius: $r = \frac{26}{2} = 13$ cm. - Half chord length: $\frac{10}{2} = 5$ cm. - Distance from center to chord: $$d = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$$ cm. 4. **Problem (d):** Calculate the diameter of a circle with a 48 cm chord 10 cm from the center. - Half chord length: $\frac{48}{2} = 24$ cm. - Distance from center to chord: $d = 10$ cm. - Use Pythagoras to find radius: $$r = \sqrt{d^2 + \left(\frac{c}{2}\right)^2} = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26$$ cm. - Diameter: $2r = 2 \times 26 = 52$ cm. 5. **Problem (e):** Calculate the distance of the midpoint of a 24 cm chord from the center of a circle with radius 13 cm. - Half chord length: $\frac{24}{2} = 12$ cm. - Distance from center to chord midpoint: $$d = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5$$ cm. **Final answers:** (a) $360\pi$ cm$^2$ (b) 3 cm (c) 12 cm (d) 52 cm (e) 5 cm