A chocolatier makes chocolate bon-bons in the shape of a sphere with a radius of 1.1 cm. The chocolate used in the bon-bons has a density of 1.28 g/cm^3 3 . If the chocolate used costs $0.03 per gram, how much would the chocolate for 190 bon-bons cost, to the nearest cent?
The Correct Answer and Explanation is :
To determine the total cost of the chocolate for 190 bon-bons, we will follow these steps:
Step 1: Calculate the Volume of One Bon-Bon
The volume of a sphere is given by the formula:
[
V = \frac{4}{3} \pi r^3
]
where ( r ) is the radius of the sphere. Given the radius ( r = 1.1 \, \text{cm} ):
[
V = \frac{4}{3} \pi (1.1)^3 = \frac{4}{3} \pi (1.331) \approx 5.575 \, \text{cm}^3
]
Step 2: Calculate the Mass of One Bon-Bon
The mass of an object is calculated using the formula:
[
\text{Mass} = \text{Volume} \times \text{Density}
]
The density of the chocolate is 1.28 g/cm³. Therefore, the mass of one bon-bon is:
[
\text{Mass} = 5.575 \, \text{cm}^3 \times 1.28 \, \text{g/cm}^3 \approx 7.136 \, \text{g}
]
Step 3: Calculate the Total Mass of 190 Bon-Bons
To find the total mass of 190 bon-bons, we multiply the mass of one bon-bon by 190:
[
\text{Total Mass} = 7.136 \, \text{g} \times 190 \approx 1355.84 \, \text{g}
]
Step 4: Calculate the Total Cost of the Chocolate
The cost of the chocolate is $0.03 per gram. So, the total cost for 1355.84 g of chocolate is:
[
\text{Total Cost} = 1355.84 \, \text{g} \times 0.03 \, \text{dollars/g} \approx 40.68 \, \text{dollars}
]
Final Answer:
The cost of the chocolate for 190 bon-bons is approximately $40.68.
Explanation:
This problem involves calculating the volume, mass, and cost of a set of chocolate bon-bons shaped like spheres. First, we use the formula for the volume of a sphere to find the volume of one bon-bon, given the radius. Then, we use the density to convert the volume into mass. Afterward, we find the total mass of 190 bon-bons. Finally, we multiply the total mass by the cost per gram of chocolate to determine the overall expense. This method allows us to precisely determine the cost based on the given parameters.