To find the volume of the solid object composed of a cone and a hemisphere, we need to calculate the volumes of both the cone and the hemisphere separately and then add them together.

Volume of the Cone

The formula for the volume of a cone is: $V_{\text{cone}} = \frac{1}{3} \pi r^2 h$

We already know the radius $r = 7 \, \text{cm}$ and the height $h = 24 \, \text{cm}$ from our previous calculations.

$V_{\text{cone}} = \frac{1}{3} \pi (7)^2 (24)$ $V_{\text{cone}} = \frac{1}{3} \pi (49) (24)$ $V_{\text{cone}} = \frac{1}{3} \pi (1176)$ $V_{\text{cone}} = 392 \pi$

Volume of the Hemisphere

The formula for the volume of a hemisphere is: $V_{\text{hemisphere}} = \frac{2}{3} \pi r^3$

We already know the radius $r = 7 \, \text{cm}$.

$V_{\text{hemisphere}} = \frac{2}{3} \pi (7)^3$ $V_{\text{hemisphere}} = \frac{2}{3} \pi (343)$ $V_{\text{hemisphere}} = \frac{2}{3} \pi (343)$ $V_{\text{hemisphere}} = \frac{686}{3} \pi$

Total Volume of the Solid

The total volume is the sum of the volumes of the cone and the hemisphere.

$V_{\text{total}} = V_{\text{cone}} + V_{\text{hemisphere}}$ $V_{\text{total}} = 392 \pi + \frac{686}{3} \pi$ $V_{\text{total}} = \pi \left( 392 + \frac{686}{3} \right)$ $V_{\text{total}} = \pi \left( \frac{1176 + 686}{3} \right)$ $V_{\text{total}} = \pi \left( \frac{1862}{3} \right)$ $V_{\text{total}} = \frac{1862}{3} \pi$

Approximating $\pi$ as 3.14:

$V_{\text{total}} \approx \frac{1862}{3} \times 3.14$ $V_{\text{total}} \approx 620.67 \times 3.14$ $V_{\text{total}} \approx 1948.90 \, \text{cm}^3$

Summary

The volume of the solid object is approximately $1948.90 \, \text{cm}^3$.

Would you like more details or have any questions?

Here are five related questions you might find useful:

1. How do you find the volume of a cylinder?
2. What is the surface area of a sphere?
3. How do you calculate the volume of a rectangular prism?
4. What is the formula for the volume of a pyramid?
5. How do you find the height of a cone if you know its volume and radius?

Tip: When working with combined solids, always handle each shape separately to ensure accuracy, then sum their respective properties as needed.