A chocolatier makes chocolate bon-bons in the shape of a sphere with a radius of 2 cm. The chocolate used in the bon-bons has a density of 1.31 g/cmº. If the chocolate used costs $0.06 per gram, how much would the chocolate for 140 bon-bons cost, to the nearest cent?
The Correct Answer and Explanation is :
To find the cost of the chocolate used in 140 bon-bons, we need to first calculate the volume of chocolate used for one bon-bon, then its mass, and finally determine the total cost for 140 bon-bons.
Step 1: Calculate the Volume of One Bon-Bon
Since the bon-bons are spherical, the formula for the volume of a sphere is:
[
V = \frac{4}{3} \pi r^3
]
where ( r ) is the radius of the sphere. For this problem, ( r = 2 \, \text{cm} ).
Substitute the radius into the formula:
[
V = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi \times 8 = \frac{32}{3} \pi \approx 33.51 \, \text{cm}^3
]
So, the volume of one bon-bon is approximately 33.51 cubic centimeters.
Step 2: Calculate the Mass of Chocolate in One Bon-Bon
The mass of the chocolate in one bon-bon can be calculated using the density of the chocolate. The formula is:
[
\text{mass} = \text{density} \times \text{volume}
]
The density of the chocolate is 1.31 g/cm³, and the volume is 33.51 cm³.
[
\text{mass} = 1.31 \, \text{g/cm}^3 \times 33.51 \, \text{cm}^3 = 43.89 \, \text{g}
]
So, the mass of chocolate in one bon-bon is approximately 43.89 grams.
Step 3: Calculate the Cost of Chocolate for One Bon-Bon
Now, we calculate the cost of the chocolate in one bon-bon. The cost is $0.06 per gram:
[
\text{cost per bon-bon} = 43.89 \, \text{g} \times 0.06 \, \text{\$/g} = 2.63 \, \text{\$}
]
So, the cost of chocolate for one bon-bon is approximately $2.63.
Step 4: Calculate the Total Cost for 140 Bon-Bons
To find the total cost for 140 bon-bons, we multiply the cost for one bon-bon by 140:
[
\text{total cost} = 2.63 \, \text{\$} \times 140 = 368.20 \, \text{\$}
]
Thus, the total cost of chocolate for 140 bon-bons is approximately $368.20.
Conclusion
The total cost for the chocolate used in 140 bon-bons is $368.20.