IDNLearn.com offers a reliable platform for finding accurate and timely answers. Find the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
Certainly! Let's break down the problem step by step:
1. Original Dimensions of the Cone:
- Height of the cone, [tex]\( h_{\text{cone}} = 16.8 \)[/tex] cm.
- Radius of the base of the cone, [tex]\( R_{\text{cone}} = 4.2 \)[/tex] cm.
2. Volume of the Cone:
The formula for the volume of a cone is given by:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi R_{\text{cone}}^2 h_{\text{cone}} \][/tex]
Substituting the given values:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi (4.2)^2 (16.8) \][/tex]
The calculated volume of the cone is approximately [tex]\( 310.34 \)[/tex] cubic cm.
3. Volume of the Sphere:
The cone is melted and recast into a sphere. Thus, the volume of the sphere, [tex]\( V_{\text{sphere}} \)[/tex], is equal to the volume of the cone:
[tex]\[ V_{\text{sphere}} = V_{\text{cone}} = 310.34 \text{ cubic cm} \][/tex]
4. Formula for the Volume of a Sphere:
The formula for the volume of a sphere is given by:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r_{\text{sphere}}^3 \][/tex]
Since [tex]\( V_{\text{sphere}} \)[/tex] is already known (310.34 cubic cm), we set up the equation:
[tex]\[ 310.34 = \frac{4}{3} \pi r_{\text{sphere}}^3 \][/tex]
5. Solving for the Radius of the Sphere:
Isolate [tex]\( r_{\text{sphere}}^3 \)[/tex]:
[tex]\[ r_{\text{sphere}}^3 = \frac{3 \times 310.34}{4 \pi} \][/tex]
Simplify the right-hand side:
[tex]\[ r_{\text{sphere}}^3 = 74.09 \][/tex]
Taking the cube root of both sides to find [tex]\( r_{\text{sphere}} \)[/tex]:
[tex]\[ r_{\text{sphere}} = \sqrt[3]{74.09} \][/tex]
The calculated radius of the sphere is approximately [tex]\( 4.2 \)[/tex] cm.
Therefore, the radius of the sphere recast from the cone is [tex]\( 4.2 \)[/tex] cm.
1. Original Dimensions of the Cone:
- Height of the cone, [tex]\( h_{\text{cone}} = 16.8 \)[/tex] cm.
- Radius of the base of the cone, [tex]\( R_{\text{cone}} = 4.2 \)[/tex] cm.
2. Volume of the Cone:
The formula for the volume of a cone is given by:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi R_{\text{cone}}^2 h_{\text{cone}} \][/tex]
Substituting the given values:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi (4.2)^2 (16.8) \][/tex]
The calculated volume of the cone is approximately [tex]\( 310.34 \)[/tex] cubic cm.
3. Volume of the Sphere:
The cone is melted and recast into a sphere. Thus, the volume of the sphere, [tex]\( V_{\text{sphere}} \)[/tex], is equal to the volume of the cone:
[tex]\[ V_{\text{sphere}} = V_{\text{cone}} = 310.34 \text{ cubic cm} \][/tex]
4. Formula for the Volume of a Sphere:
The formula for the volume of a sphere is given by:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r_{\text{sphere}}^3 \][/tex]
Since [tex]\( V_{\text{sphere}} \)[/tex] is already known (310.34 cubic cm), we set up the equation:
[tex]\[ 310.34 = \frac{4}{3} \pi r_{\text{sphere}}^3 \][/tex]
5. Solving for the Radius of the Sphere:
Isolate [tex]\( r_{\text{sphere}}^3 \)[/tex]:
[tex]\[ r_{\text{sphere}}^3 = \frac{3 \times 310.34}{4 \pi} \][/tex]
Simplify the right-hand side:
[tex]\[ r_{\text{sphere}}^3 = 74.09 \][/tex]
Taking the cube root of both sides to find [tex]\( r_{\text{sphere}} \)[/tex]:
[tex]\[ r_{\text{sphere}} = \sqrt[3]{74.09} \][/tex]
The calculated radius of the sphere is approximately [tex]\( 4.2 \)[/tex] cm.
Therefore, the radius of the sphere recast from the cone is [tex]\( 4.2 \)[/tex] cm.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.