Final answer:
To fill a spherical balloon with a surface area of 900 cm², about 2.558 liters of helium are needed. For 500 helium-filled balloons, the total helium cost would be $30.702, and the total balloon cost would be $35.00, leading to a grand total of $65.702.
Step-by-step explanation:
The student is asking how much helium is needed to fill a spherical balloon with a surface area of 900 cm2 and the total cost of 500 such helium-filled balloons. The surface area (A) of a sphere can be expressed as A = 4πr2, where r is the radius of the sphere. To find the volume of helium required, we need to solve for the radius, calculate the volume and then consider the cost of helium and balloons.
First, we calculate the radius of the balloon using the surface area formula:
- A = 900 cm2 = 4πr2
- r2 = A / (4π)
- r2 = 900 / (4π)
- r = √(900 / (4π))
- r ≈ 8.485 cm
Then we calculate the volume (V) of the balloon using the formula for the volume of a sphere:
- V = (4/3)πr3
- V = (4/3)π(8.485)3
- V ≈ 2558.55 cm3
- V ≈ 2.558 L (since 1 L = 1000 cm3)
Calculating the cost for 500 helium-filled balloons:
- Total volume for 500 balloons = 500 × 2.558 L = 1279.25 L
- Total helium cost = 1279.25 L × $0.024/L = $30.702
- Total balloon cost = 500 × $0.07 = $35.00
- Grand total = Total helium cost + Total balloon cost = $30.702 + $35.00 = $65.702
Hence, it would cost $65.702 to fill and purchase 500 helium-filled balloons.