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
Answers
GPT-4o mini
To find the volume of the creamy center inside the sphere-shaped piece of chocolate, we'll first use the formula for the volume of a sphere, which is given by:
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Since the chocolate piece has a diameter of 1 inch, the radius \( r \) is:
\[
r = \frac{\text{diameter}}{2} = \frac{1}{2} \text{ inches} = 0.5 \text{ inches}
\]
Now we can substitute the radius into the volume formula:
\[
V = \frac{4}{3} \pi (0.5)^3
\]
Calculating \( (0.5)^3 \):
\[
(0.5)^3 = 0.125
\]
Now substituting this back into the volume formula:
\[
V = \frac{4}{3} \pi (0.125) = \frac{4 \times 0.125}{3} \pi = \frac{0.5}{3} \pi = \frac{1}{6} \pi
\]
To express the volume in terms of π:
\[
V \approx \frac{1}{6} \pi
\]
Now, to also find the numerical value of \( \frac{1}{6} \pi \) rounded to the nearest tenth:
Using \( \pi \approx 3.14 \):
\[
\frac{1}{6} \pi \approx \frac{3.14}{6} \approx 0.5233
\]
Rounded to the nearest tenth, this value is:
\[
0.5
\]
So the volume of the creamy filling that can fit inside the chocolate piece is:
\[
\frac{1}{6} \pi \approx 0.5 \text{ inches}^3
\]
Final rounded value:
\[
\text{Volume of creamy filling} \approx 0.5 \text{ inches}^3
\]
\[
V = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.
Since the chocolate piece has a diameter of 1 inch, the radius \( r \) is:
\[
r = \frac{\text{diameter}}{2} = \frac{1}{2} \text{ inches} = 0.5 \text{ inches}
\]
Now we can substitute the radius into the volume formula:
\[
V = \frac{4}{3} \pi (0.5)^3
\]
Calculating \( (0.5)^3 \):
\[
(0.5)^3 = 0.125
\]
Now substituting this back into the volume formula:
\[
V = \frac{4}{3} \pi (0.125) = \frac{4 \times 0.125}{3} \pi = \frac{0.5}{3} \pi = \frac{1}{6} \pi
\]
To express the volume in terms of π:
\[
V \approx \frac{1}{6} \pi
\]
Now, to also find the numerical value of \( \frac{1}{6} \pi \) rounded to the nearest tenth:
Using \( \pi \approx 3.14 \):
\[
\frac{1}{6} \pi \approx \frac{3.14}{6} \approx 0.5233
\]
Rounded to the nearest tenth, this value is:
\[
0.5
\]
So the volume of the creamy filling that can fit inside the chocolate piece is:
\[
\frac{1}{6} \pi \approx 0.5 \text{ inches}^3
\]
Final rounded value:
\[
\text{Volume of creamy filling} \approx 0.5 \text{ inches}^3
\]