Certainly! Let's address the problem step-by-step and find the required volumes and radius.
### Part (a)
The formula to find the volume [tex]\(V\)[/tex] of a sphere when the radius [tex]\(r\)[/tex] is given is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
### Part (b)
We have three metallic spheres with radii [tex]\(3 \text{ cm}, 4 \text{ cm},\)[/tex] and [tex]\(5 \text{ cm}\)[/tex]. We will first find the volumes of each of these spheres.
1. Volume of the sphere with radius [tex]\(3 \text{ cm}\)[/tex]:
[tex]\[
V_1 = \frac{4}{3} \pi (3)^3 = 113.097 \text{ cubic centimeters}
\][/tex]
2. Volume of the sphere with radius [tex]\(4 \text{ cm}\)[/tex]:
[tex]\[
V_2 = \frac{4}{3} \pi (4)^3 = 268.083 \text{ cubic centimeters}
\][/tex]
3. Volume of the sphere with radius [tex]\(5 \text{ cm}\)[/tex]:
[tex]\[
V_3 = \frac{4}{3} \pi (5)^3 = 523.599 \text{ cubic centimeters}
\][/tex]
Next, we will find the total volume of the new single sphere formed by melting these three spheres.
Total volume [tex]\(V_{total}\)[/tex]:
[tex]\[
V_{total} = V_1 + V_2 + V_3 = 113.097 + 268.083 + 523.599 = 904.779 \text{ cubic centimeters}
\][/tex]
To find the radius [tex]\(R\)[/tex] of the new single sphere, we use the inverse formula derived from the volume formula:
[tex]\[
V = \frac{4}{3} \pi R^3 \implies R^3 = \frac{3V}{4\pi}
\][/tex]
Substitute [tex]\(V_{total}\)[/tex]:
[tex]\[
R = \left( \frac{3 \times 904.779}{4 \pi} \right)^{\frac{1}{3}} = 6 \text{ cm}
\][/tex]
Therefore:
- The total volume of the new sphere is [tex]\(904.779 \text{ cubic centimeters}\)[/tex].
- The radius of the new sphere is [tex]\(6 \text{ cm}\)[/tex].
