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The radius of a sphere, [tex]\( r \)[/tex], is given by the formula below, where [tex]\( s \)[/tex] is the surface area of the sphere.
[tex]\[ r = \frac{1}{2} \sqrt{\frac{s}{\pi}} \][/tex]

A spherical balloon has a maximum surface area of 1,500 square centimeters.

Use the given formula to write a function, [tex]\( r(s) \)[/tex], that models the situation. Then, use the function to predict how the radius of the balloon changes as the balloon is inflated.

A. As the surface area of the balloon increases, the radius of the balloon increases until the maximum surface area is reached.

B. As the surface area of the balloon increases, the radius of the balloon increases without bound.

C. As the surface area of the balloon increases, the radius of the balloon decreases without bound.

D. As the surface area of the balloon increases, the radius of the balloon decreases until the maximum surface area is reached.


Sagot :

Let's begin by interpreting and deriving the function [tex]\(r(s)\)[/tex] from the given problem statement. The given formula is:

[tex]\[ f = \frac{1}{2} \sqrt{\frac{T}{\pi}} \][/tex]

We need to model the function [tex]\(r(s)\)[/tex] where [tex]\(s\)[/tex] is the surface area of a spherical balloon. In this context, [tex]\(T = s\)[/tex], so the formula translates into:

[tex]\[ r(s) = \frac{1}{2} \sqrt{\frac{s}{\pi}} \][/tex]

This function [tex]\(r(s)\)[/tex] gives us the radius of the balloon as a function of its surface area [tex]\(s\)[/tex].

Next, we predict how the radius of the balloon changes as the balloon is inflated, given that the maximum surface area is 1,500 square centimeters.

1. For very small surface area [tex]\(s \approx 0\)[/tex]:
[tex]\[ r(0) = \frac{1}{2} \sqrt{\frac{0}{\pi}} = 0 \][/tex]
When the surface area is very small (close to zero), the radius is also close to zero.

2. For the maximum surface area of [tex]\(s = 1500\)[/tex] square centimeters:
[tex]\[ r(1500) = \frac{1}{2} \sqrt{\frac{1500}{\pi}} = \frac{1}{2} \sqrt{\frac{1500}{3.14159265359}} = 10.925484305920792 \][/tex]
So, when the surface area reaches its maximum of 1500 square centimeters, the radius is approximately 10.93 cm.

Now, let's summarize the behavior of the radius as the surface area increases:

- When the surface area is zero, the radius is zero.
- As the surface area increases, the radius increases accordingly.
- When the surface area is at its maximum (1500 square centimeters), the radius reaches approximately 10.93 cm.
- There is no indication that the radius will decrease as the surface area increases.

Given the above analysis, the correct option is:
[tex]\[ \boxed{\text{A. As the surface area of the balloon increases, the radius of the balloon increases until the maximum surface area is reached.}} \][/tex]