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Two spheres have surface areas of [tex]100 \pi[/tex] units[tex]\(^2\)[/tex] and [tex]36 \pi[/tex] units[tex]\(^2\)[/tex].

If the volume of the larger sphere is [tex]\frac{500}{3} \pi[/tex] units[tex]\(^3\)[/tex], determine the following measures:

- Scale factor = [tex]\(\square\)[/tex]
- Radius of smaller sphere = [tex]\(\square\)[/tex] units
- Radius of larger sphere = [tex]\(\square\)[/tex] units
- Volume of smaller sphere = [tex]\(\square \pi\)[/tex] units[tex]\(^3\)[/tex]

Sagot :

GinnyAnswer
Let's break down the problem step-by-step.

1. Finding the radii of the spheres:

Given surface areas:
- Surface area of the larger sphere: [tex]\( 100 \pi \)[/tex] units²
- Surface area of the smaller sphere: [tex]\( 36 \pi \)[/tex] units²

The formula for the surface area of a sphere is:
[tex]\[ 4\pi r^2 \][/tex]

From this, we can solve for the radius [tex]\( r \)[/tex] of each sphere.

For the larger sphere:
[tex]\[ 4\pi r_{\text{large}}^2 = 100 \pi \][/tex]
[tex]\[ r_{\text{large}}^2 = \frac{100 \pi}{4 \pi} = 25 \][/tex]
[tex]\[ r_{\text{large}} = \sqrt{25} = 5 \text{ units} \][/tex]

For the smaller sphere:
[tex]\[ 4\pi r_{\text{small}}^2 = 36 \pi \][/tex]
[tex]\[ r_{\text{small}}^2 = \frac{36 \pi}{4 \pi} = 9 \][/tex]
[tex]\[ r_{\text{small}} = \sqrt{9} = 3 \text{ units} \][/tex]

2. Determining the scale factor:

The scale factor is the ratio of the radii of the larger sphere to the smaller sphere.
[tex]\[ \text{Scale factor} = \frac{r_{\text{large}}}{r_{\text{small}}} = \frac{5}{3} \approx 1.667 \][/tex]

3. Finding the volume of the smaller sphere:

Given the volume of the larger sphere:
[tex]\[ \frac{500}{3} \pi \text{ units}^3 \][/tex]

We know the formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]

For the smaller sphere, using its radius [tex]\( r_{\text{small}} = 3 \)[/tex]:
[tex]\[ V_{\text{small}} = \frac{4}{3} \pi r_{\text{small}}^3 \][/tex]
[tex]\[ V_{\text{small}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ units}^3 \][/tex]

So, summarizing the results:
- Radius of the larger sphere: [tex]\( 5 \)[/tex] units
- Radius of the smaller sphere: [tex]\( 3 \)[/tex] units
- Scale factor: [tex]\( 1.667 \)[/tex]
- Volume of the smaller sphere: [tex]\( \approx 113.097 \)[/tex] units³

These are the complete answers required.
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