Alright, let's go through each part of the problem step-by-step:
### Part (a): Finding the Slant Height of the Cone
Given:
- Height of the cone ([tex]\( h \)[/tex]) = 24 cm
- Diameter of the base = 14 cm
- Therefore, the radius of the base ([tex]\( r \)[/tex]) = 14 / 2 = 7 cm
To find the slant height ([tex]\( l \)[/tex]) of the cone, we use the Pythagorean theorem:
[tex]\[ l = \sqrt{r^2 + h^2} \][/tex]
Substitute the values:
[tex]\[ l = \sqrt{7^2 + 24^2} \][/tex]
[tex]\[ l = \sqrt{49 + 576} \][/tex]
[tex]\[ l = \sqrt{625} \][/tex]
[tex]\[ l = 25 \text{ cm} \][/tex]
So, the slant height of the cone is 25 cm.
### Part (b): Finding the Volume of the Solid
The solid consists of a cone and a hemisphere.
1. Volume of the Cone:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \][/tex]
Substitute the values:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi (7^2) (24) \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi (49) (24) \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi (1176) \][/tex]
[tex]\[ V_{\text{cone}} \approx 1231.50 \text{ cm}^3 \][/tex]
2. Volume of the Hemisphere:
[tex]\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \][/tex]
Substitute the values:
[tex]\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (7^3) \][/tex]
[tex]\[ V_{\text{hemisphere}} = \frac{2}{3} \pi (343) \][/tex]
[tex]\[ V_{\text{hemisphere}} \approx 718.38 \text{ cm}^3 \][/tex]
3. Total Volume of the Solid:
[tex]\[ V_{\text{solid}} = V_{\text{cone}} + V_{\text{hemisphere}} \][/tex]
[tex]\[ V_{\text{solid}} \approx 1231.50 \text{ cm}^3 + 718.38 \text{ cm}^3 \][/tex]
[tex]\[ V_{\text{solid}} \approx 1949.88 \text{ cm}^3 \][/tex]
So, the total volume of the solid is 1949.88 cm³.
### Part (c): Finding the Height of the Cylinder
When the solid is melted and reformed into a cylinder with a radius of 7 cm, we use the total volume calculated above to find the height of the cylinder.
Given:
- Volume of the cylinder ([tex]\( V_{\text{cyl}} \)[/tex]) = Volume of the solid
- Radius of the cylinder ([tex]\( r_{\text{cyl}} \)[/tex]) = 7 cm
[tex]\[ V_{\text{cyl}} = \pi r_{\text{cyl}}^2 h_{\text{cyl}} \][/tex]
Substitute the values:
[tex]\[ 1949.88 = \pi (7^2) h_{\text{cyl}} \][/tex]
[tex]\[ 1949.88 = \pi (49) h_{\text{cyl}} \][/tex]
[tex]\[ h_{\text{cyl}} = \frac{1949.88}{\pi (49)} \][/tex]
[tex]\[ h_{\text{cyl}} \approx 12.67 \text{ cm} \][/tex]
So, the height of the cylinder is approximately 12.67 cm.
