A) A cone has a radius of 3m and a height of 4m. Calculate its surface area and volume. b) A sphere has a radius of 3m. Finds its Surface area and volume.

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13. a) A cone has a radius of 3m and a height of 4m. Calculate its surface area and volume.

    • Step 1: Kokotoa urefu wa slant ($l$) kwa kutumia Theorem ya Pythagoras. $$l = \sqrt{r^2 + h^2}$$ $$l = \sqrt{(3 \text{ m})^2 + (4 \text{ m})^2}$$ $$l = \sqrt{9 \text{ m}^2 + 16 \text{ m}^2}$$ $$l = \sqrt{25 \text{ m}^2}$$ $$l = 5 \text{ m}$$
    • Step 2: Kokotoa eneo la uso ($A$) la koni. Fomula ni jumla ya eneo la duara la chini na eneo la uso wa kando. $$A = \pi r^2 + \pi r l$$ $$A = \pi (3 \text{ m})^2 + \pi (3 \text{ m})(5 \text{ m})$$ $$A = 9\pi \text{ m}^2 + 15\pi \text{ m}^2$$ $$A = 24\pi \text{ m}^2$$
    • Step 3: Kokotoa ujazo ($V$) wa koni. $$V = \frac{1}{3}\pi r^2 h$$ $$V = \frac{1}{3}\pi (3 \text{ m})^2 (4 \text{ m})$$ $$V = \frac{1}{3}\pi (9 \text{ m}^2) (4 \text{ m})$$ $$V = 12\pi \text{ m}^3$$
    • Jibu la Mwisho: $$\boxed{\mathbf{\text{Eneo la uso} = 24\pi \text{ m}^2, \text{Ujazo} = 12\pi \text{ m}^3}}$$

b) A sphere has a radius of 3m. Finds its Surface area and volume.

*   **Step 1:** Kokotoa eneo la uso ($A$) la tufe.
    $$
    A = 4\pi r^2
    $$
    $$
    A = 4\pi (3 \text{ m})^2
    $$
    $$
    A = 4\pi (9 \text{ m}^2)
    $$
    $$
    A = 36\pi \text{ m}^2
    $$
*   **Step 2:** Kokotoa ujazo ($V$) wa tufe.
    $$
    V = \frac{4}{3}\pi r^3
    $$
    $$
    V = \frac{4}{3}\pi (3 \text{ m})^3
    $$
    $$
    V = \frac{4}{3}\pi (27 \text{ m}^3)
    $$
    $$
    V = 4\pi (9 \text{ m}^3)
    $$
    $$
    V = 36\pi \text{ m}^3
    $$
*   **Jibu la Mwisho:**
    $$
    \boxed{\mathbf{\text{Eneo la uso} = 36\pi \text{ m}^2, \text{Ujazo} = 36\pi \text{ m}^3}}
    $$