Answered

Three metallic spheres of radii [tex]3 \, \text{cm}[/tex], [tex]4 \, \text{cm}[/tex], and [tex]5 \, \text{cm}[/tex] are melted and reformed into a single sphere.

a) Write the formula to find the volume of a sphere when the radius is given.

b) Find the volume and the radius of the new sphere.



Answer :

GinnyAnswer
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].

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