A design on the surface of a balloon is 5 cm wide when the balloon holds 71 cm^3 of air. How much air does the balloon hold when the design is 10 cm wide? Explain the method you use to find the amount of air.
How would I solve this step by step?
the volume would be proportional to the cube of the linear measurements
Since you doubled the 5 cm to 10 cm, the volume would be 2^3 or 8 times as large.
To solve this problem, we can use the concept of proportional relationships. Let's go through the steps:
Step 1: Identify the given information:
- The design on the surface of the balloon is 5 cm wide.
- When the balloon holds this design, it contains 71 cm^3 of air.
Step 2: Set up a proportion:
Since there is a proportional relationship between the width of the design and the volume of the air, we can set up a proportion using these two variables. Let's call the unknown volume of air when the design is 10 cm wide as "x."
Using the given information, we can set up the proportion:
5 cm / 71 cm^3 = 10 cm / x
Step 3: Solve the proportion:
To solve the proportion, we can use cross-multiplication. Multiply 5 cm by x and 71 cm^3 by 10 cm:
5cm * x = 71cm^3 * 10cm
Step 4: Simplify the equation:
On the left side of the equation, 5cm * x becomes 5x.
On the right side of the equation, 71cm^3 * 10cm becomes 710cm^4.
Thus, our equation becomes:
5x = 710cm^4
Step 5: Solve for x:
Divide both sides of the equation by 5 to isolate x:
x = 710cm^4 / 5
Step 6: Simplify the answer:
Dividing 710cm^4 by 5 gives us:
x = 142cm^4.
So when the design is 10 cm wide, the balloon would hold 142 cm^3 of air.
The method used in this solution is based on the concept of proportional relationships, which allows us to find the unknown value by setting up and solving a proportion.