A spherical balloon with a surface area of 36pi square inches is inflated further until its diameter has increased by 50%. The surface area of the balloon is now? I know the formula for surface area is

A=4pi*r squared. I don't understand how you can figure the problem without knowing what the radius is? I thought you might take half of 36 and add the answer to 36 giving me 54pi square inches.

the surface was 4 pi r^2 = 36pi

Now, replace r with (3/2) r and the new surface area is

4pi(3/2 r)^2 = 4pi * 9/4 r^2
= 4pi r^2 (9/4)
= 36pi(9/4)
= 81pi

Or, you could just solve for r and get that r increased from 3 to 9/2, so the area increased from 36pi to 81pi.

The surface area is proportional to the square of their radii

(1.5r)^2/r^2 = x/36π
2.25/1 = x/36π
x = 2.2(36π) = 81π

To find the new surface area of the balloon after it is inflated, we need to calculate the radius first.

We know the formula for the surface area of a sphere is A = 4πr^2, where A is the surface area and r is the radius.

Let's assume the initial radius of the balloon is r0 and the initial surface area is 36π square inches.

A = 4πr0^2 ------(1)

Given that the diameter of the balloon is increased by 50%, the new diameter is 1.5 times the initial diameter.

So, the new diameter = 1.5 * 2r0 = 3r0

The new radius of the balloon, r1, is half of the new diameter, so:

r1 = (3r0)/2 = 1.5r0

Now, we can calculate the new surface area of the balloon, A1:

A1 = 4πr1^2 ------(2)

Substituting the value of r1 in equation (2):

A1 = 4π(1.5r0)^2 = 4π(2.25r0^2) = 9πr0^2

So, the new surface area of the balloon is 9 times the initial surface area.

Therefore, the new surface area is 9 * 36π = 324π square inches.

Hence, the surface area of the balloon after it is inflated further is 324π square inches.

To solve this problem, we need to break it down into steps. Let's go through it together:

Step 1: Find the radius of the original spherical balloon.
The formula for the surface area of a sphere is A = 4πr^2. In this case, we are given that the surface area is 36π square inches. By rearranging the formula, we have:
36π = 4πr^2
Dividing both sides of the equation by 4π:
9 = r^2
Taking the square root of both sides, we find:
r = 3 inches

Step 2: Determine the new diameter of the balloon after it has increased by 50%.
Since the original radius was 3 inches, the original diameter was 2r = 2 * 3 = 6 inches. To find the new diameter, we need to increase it by 50%. We can simply calculate 50% of the original diameter and add it to the original diameter:
New Diameter = 6 + (0.50 * 6) = 6 + 3 = 9 inches

Step 3: Calculate the new radius using the new diameter.
To find the new radius, divide the new diameter by 2:
New Radius = 9 / 2 = 4.5 inches

Step 4: Determine the new surface area of the balloon.
We can now use the formula for the surface area of a sphere (A = 4πr^2) and substitute the value we found for the new radius:
New Surface Area = 4π(4.5)^2 = 4π(20.25) = 81π square inches

So, the surface area of the balloon after it has been inflated and its diameter increased by 50% is 81π square inches, not 54π square inches.