Question
You buy a sphere-shaped piece of chocolate that has a creamy center. The chocolate piece measures 1 inch in diameter. How much creamy filling can fit inside the chocolate piece? Calculate your answer in terms of π
and round to the nearest tenth.(1 point)
inches3
and round to the nearest tenth.(1 point)
inches3
Answers
GPT-4o mini
To find the volume of the creamy filling inside the chocolate piece, we will calculate the volume of the sphere using the formula for the volume of a sphere:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Given that the diameter of the chocolate piece is 1 inch, we can find the radius by dividing the diameter by 2:
\[
r = \frac{1}{2} \text{ inches} = 0.5 \text{ inches}
\]
Now, we will substitute this value of \( r \) into the volume formula:
\[
V = \frac{4}{3} \pi (0.5)^3
\]
Calculating \( (0.5)^3 \):
\[
(0.5)^3 = 0.125
\]
Now substitute this back into the volume formula:
\[
V = \frac{4}{3} \pi (0.125)
\]
Calculating \( \frac{4}{3} \times 0.125 \):
\[
\frac{4}{3} \times 0.125 = \frac{4 \times 0.125}{3} = \frac{0.5}{3} \approx 0.1667
\]
Thus, the volume is:
\[
V = \frac{4}{3} \times 0.125 \pi \approx 0.1667 \pi
\]
To round this value, we convert it to a decimal:
\[
0.1667 \pi \approx (0.1667 \times 3.14159) \approx 0.524
\]
Rounding 0.524 to the nearest tenth gives us:
\[
0.5
\]
Therefore, the volume of the creamy filling that can fit inside the chocolate piece is approximately \( 0.5 \) inches³.
To summarize:
\[
\text{Volume in terms of } \pi: \frac{1}{6} \pi \, \text{ cubic inches} \quad \text{(approximately 0.5 rounded to the nearest tenth)}
\]
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Given that the diameter of the chocolate piece is 1 inch, we can find the radius by dividing the diameter by 2:
\[
r = \frac{1}{2} \text{ inches} = 0.5 \text{ inches}
\]
Now, we will substitute this value of \( r \) into the volume formula:
\[
V = \frac{4}{3} \pi (0.5)^3
\]
Calculating \( (0.5)^3 \):
\[
(0.5)^3 = 0.125
\]
Now substitute this back into the volume formula:
\[
V = \frac{4}{3} \pi (0.125)
\]
Calculating \( \frac{4}{3} \times 0.125 \):
\[
\frac{4}{3} \times 0.125 = \frac{4 \times 0.125}{3} = \frac{0.5}{3} \approx 0.1667
\]
Thus, the volume is:
\[
V = \frac{4}{3} \times 0.125 \pi \approx 0.1667 \pi
\]
To round this value, we convert it to a decimal:
\[
0.1667 \pi \approx (0.1667 \times 3.14159) \approx 0.524
\]
Rounding 0.524 to the nearest tenth gives us:
\[
0.5
\]
Therefore, the volume of the creamy filling that can fit inside the chocolate piece is approximately \( 0.5 \) inches³.
To summarize:
\[
\text{Volume in terms of } \pi: \frac{1}{6} \pi \, \text{ cubic inches} \quad \text{(approximately 0.5 rounded to the nearest tenth)}
\]