A spherical baloon expands when it is taken from the cold outdors to the inside of a warm house. if its suface area increases 16%, by what percentage does the radius change

As = 4pi*r^2 = Surface area.

100% + 16% = 116% = 1.16
A2/A1 = 4pi*r2^2/4pi*r1^2 = 1.16
The 4pi cancels:
r2^2 / r1^2 = 1.16
Take sq. root of both sides:
r2/r1 = 1.0770 = 107.7%
%Change = 107.7% - 100% = 7.7% Increase.

The surface area is proportional to
the square of the radius:

7.7%

Therefore, the radius increases by approximately 7.7%.

To find the percentage change in radius, we first need to understand the relationship between the surface area and the radius of a sphere.

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

Surface area = 4πr^2

where "r" represents the radius of the sphere.

Now, let's say the initial radius of the balloon is "r" and after it expands, the new radius becomes "r + Δr", where Δr represents the change in radius.

We are given that the surface area increases by 16%. This means the new surface area is 116% of the initial surface area.

So, we can set up the equation:

New surface area = Initial surface area + (16% of Initial surface area)

4π(r + Δr)^2 = 4πr^2 + (0.16 * 4πr^2)

Now, we can simplify the equation:

(r + Δr)^2 = r^2 + (0.16 * r^2)

Expanding the equation and simplifying:

r^2 + 2rΔr + Δr^2 = r^2 + 0.16r^2

Subtracting r^2 from both sides:

2rΔr + Δr^2 = 0.16r^2

Dividing both sides by r:

2Δr + Δr^2/r = 0.16r

As Δr/r is relatively small, we can approximate it to zero, neglecting Δr^2/r. This allows us to simplify the equation further:

2Δr = 0.16r

Dividing both sides by 2:

Δr = 0.16r/2

Simplifying:

Δr = 0.08r

To find the percentage change in radius, we can now divide Δr by the initial radius and multiply by 100:

Percentage change in radius = (Δr / r) * 100

Percentage change in radius = (0.08r / r) * 100

Percentage change in radius = 8%

Therefore, the radius changes by 8%.