The radius of the base of a right circular cone and its height are in the ratio 5:12. If the volume of the cone is 2512 cm³, then its oblique height is:

Question

Accepted Answer

To determine why Option 3 (26 cm) is the correct answer for the oblique height of the cone, we need to follow these steps:

1. **Understanding the Problem:**
   - We are given that the radius $ r $ and height $ h $ of the cone are in the ratio 5:12.
   - The volume of the cone is 2512 cm³.
   - We need to find the oblique height (slant height) of the cone.

2. **Using the Volume Formula:**
   - The volume $ V $ of a cone is given by the formula:
     $$
     V = \frac{1}{3} \pi r^2 h
     $$
   - Substituting the given volume:
     $$
     \frac{1}{3} \pi r^2 h = 2512
     $$

3. **Expressing Radius and Height in Terms of a Common Variable:**
   - Let the common ratio factor be $ k $.
   - Then, $ r = 5k $ and $ h = 12k $.

4. **Substituting into the Volume Formula:**
   - Substitute $ r = 5k $ and $ h = 12k $ into the volume formula:
     $$
     \frac{1}{3} \pi (5k)^2 (12k) = 2512
     $$
   - Simplifying:
     $$
     \frac{1}{3} \pi \times 25k^2 \times 12k = 2512
     $$
     $$
     100\pi k^3 = 2512
     $$
     $$
     k^3 = \frac{2512}{100\pi}
     $$
     $$
     k^3 \approx \frac{2512}{314.16} \approx 8
     $$
     $$
     k \approx 2
     $$

5. **Calculating the Radius and Height:**
   - $ r = 5k = 5 \times 2 = 10 $ cm
   - $ h = 12k = 12 \times 2 = 24 $ cm

6. **Finding the Oblique Height:**
   - The oblique height $ l $ (slant height) of the cone can be found using the Pythagorean theorem:
     $$
     l = \sqrt{r^2 + h^2}
     $$
     $$
     l = \sqrt{10^2 + 24^2}
     $$
     $$
     l = \sqrt{100 + 576}
     $$
     $$
     l = \sqrt{676}
     $$
     $$
     l = 26 \text{ cm}
     $$

7. **Conclusion:**
   - Therefore, the oblique height of the cone is 26 cm, which corresponds to Option 3.

**Explanation for Other Options:**
- **Option 1 (22 cm):** This value is incorrect because the calculation of the slant height using the Pythagorean theorem results in 26 cm, not 22 cm.
- **Option 2 (24 cm):** This is the vertical height of the cone, not the slant height. The slant height is longer due to the Pythagorean relationship.
- **Option 4 (28 cm):** This value exceeds the calculated slant height of 26 cm, making it incorrect.

Thus, Option 3 is the correct answer, as it accurately reflects the calculated slant height of the cone.