A spherical balloon expands when it is taken from the cold outdoors into a warm house. If the surface area increases by 50% (factor of 1.50), by what percentage does the radius of the balloon change?

a)7.7%
b)50%
c)100%
d)75%
e)22%

radius is proportional the square root of area.

R2/R1 = sqrt(A2/A1)

relaive change = (R2 - R1)/R1
= (R2/R1) - 1

sqrt(1.50) - 1 = 0.2247 = 22.47%

To find the percentage change in the radius of the balloon, we can use the formula:

\(\text{{Percentage change in radius}} = \left(\frac{{\text{{New radius}} - \text{{Initial radius}}}}{{\text{{Initial radius}}}}\right) \times 100\)

We are given that the surface area of the balloon increases by a factor of 1.50, which means the new surface area is 1.50 times the initial surface area. Since the surface area of a sphere is directly proportional to the square of its radius, we can write:

\(\frac{{\text{{New surface area}}}}{{\text{{Initial surface area}}}} = \left(\frac{{\text{{New radius}}}}{{\text{{Initial radius}}}}\right)^2\)

Substituting the factor of 1.50 for the ratio of surface areas, we have:

\(1.50 = \left(\frac{{\text{{New radius}}}}{{\text{{Initial radius}}}}\right)^2\)

Taking the square root of both sides:

\(\sqrt{1.50} = \frac{{\text{{New radius}}}}{{\text{{Initial radius}}}}\)

Now, let's find the value of \(\sqrt{1.50}\):

\(\sqrt{1.50} \approx 1.22474\)

Therefore, the new radius is approximately 1.22474 times the initial radius. To find the percentage change in the radius, we can use the percentage change formula:

\(\text{{Percentage change in radius}} = (1.22474 - 1) \times 100\)

\(\text{{Percentage change in radius}} = 0.22474 \times 100\)

\(\text{{Percentage change in radius}} \approx 22.474\%\)

Therefore, the correct answer is e) 22%.

To figure out the percentage change in the radius of the balloon, we need to utilize the relationship between the surface area of a sphere and its radius.

The surface area of a sphere is given by the formula:

Surface Area = 4πr^2

Where r represents the radius of the sphere.

Since we are given that the surface area increases by 50%, we can set up the following equation:

New Surface Area = (Original Surface Area) + (50% of Original Surface Area)
New Surface Area = Original Surface Area + (0.5 * Original Surface Area)

Simplifying this equation, we have:

New Surface Area = 1.5 * Original Surface Area

Now, we can equate this to the formula for the surface area of a sphere:

4πr^2 = 1.5 * 4πr^2

Simplifying by canceling out the common terms (4πr^2), we get:

1 = 1.5

This implies that there is no solution, meaning the equation does not hold true.

Therefore, there is an error in the problem statement. Without the correct information, we cannot determine the percentage change in the radius of the balloon.