Question

The height of a balloon, h, varies directly as the square root of its surface area, A.

When the balloon's surface area is 81 its height is 12
What is its height when its surface area is 144?

Answer 1

To find the balloon's height for a surface area of 144, we first determine the constant of proportionality from the given information (A=81, h=12) and then apply it to calculate the new height, resulting in a height of 16 units.

Step-by-step explanation:

The student is working on a problem where the height of a balloon h is directly proportional to the square root of its surface area A. We're given that when A = 81, the height h = 12, and we need to find the new height when A = 144.

We can set up a proportion because the height varies directly as the square root of the surface area. From the given information:

h = k √A (where k is the constant of proportionality)

12 = k √81

k = 12 / 9

k = 4/3

Now we can find the height for A = 144:

h = (4/3) √144

h = (4/3) * 12

h = 16

Therefore, when the surface area of the balloon is 144, its height is 16 units.

Answer 2

The relationship between height (h) and the square root of the surface area (A) is: h = (4/3) * √A.

Certainly! Let's break down the relationship between the height of the balloon (h) and its surface area (A), given the information provided:

Direct Variation: The relationship between the height of the balloon and the square root of its surface area is described as a direct variation. This means that as one quantity (height) changes, the other (square root of surface area) changes proportionally.

Given Information: When the balloon's surface area is 81, its height is 12. Mathematically, this is represented as A = 81 and h = 12.

Using the Square Root: The square root of the surface area (A = 81) is √81 = 9. Therefore, the square root of the surface area is 9.

Setting up the Proportion: Using the given information (h = 12 when A = 81), the direct variation equation is: h = k * √A, where k is the constant of variation.

Solving for k: Substituting the values (h = 12 and √A = 9), the equation becomes 12 = k * 9.

Calculating k: To find k, divide both sides by 9: k = 12 / 9 = 4/3.

Final Equation: The relationship between height (h) and the square root of the surface area (A) is: h = (4/3) * √A.

Complete Question:

The height of a balloon, h, varies directly as the square root of its surface area, A.

When the balloon's surface area is 81 its height is 12