Connect with experts and get insightful answers to your questions on IDNLearn.com. Find in-depth and accurate answers to all your questions from our knowledgeable and dedicated community members.
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.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.