Question
An autographed baseball is encased in a plastic case. The owner would like to completely fill the rest of the container with an acrylic epoxy to completely preserve the baseball. The interior dimensions of the case are 3 in. by 3 in. by 3 in. The cube perfectly inscribes the ball. How many fluid ounces of acrylic will need to be poured in to fill the remaining space? (1 cubic inch = 0.554 fluid ounces)
image from problem 7 on main assignment
Question 1 options:
7.13 fl oz
15.02 fl oz
3.6 fl oz
20.25 fl oz
image from problem 7 on main assignment
Question 1 options:
7.13 fl oz
15.02 fl oz
3.6 fl oz
20.25 fl oz
Answers
Answered by
GPT-4o mini
To find the volume of acrylic epoxy needed to fill the remaining space in the plastic case, we need to first calculate the volume of the cube and then the volume of the baseball.
1. **Calculate the volume of the cube**:
\[
\text{Volume of cube} = \text{side}^3 = 3 \, \text{in} \times 3 \, \text{in} \times 3 \, \text{in} = 27 \, \text{cubic inches}
\]
2. **Calculate the volume of the baseball**:
The baseball is a sphere that perfectly inscribes within the cube. The diameter of the baseball is equal to the side length of the cube, which is 3 inches. Therefore, the radius \( r \) of the baseball is:
\[
r = \frac{\text{diameter}}{2} = \frac{3 \, \text{in}}{2} = 1.5 \, \text{in}
\]
The volume of a sphere is calculated using the formula:
\[
\text{Volume of sphere} = \frac{4}{3} \pi r^3
\]
Substituting the value of \( r \):
\[
\text{Volume of sphere} = \frac{4}{3} \pi (1.5)^3 = \frac{4}{3} \pi (3.375) = \frac{4 \cdot 3.375 \pi}{3} = 4.5 \pi \approx 14.137 \, \text{cubic inches}
\]
3. **Calculate the volume of the acrylic epoxy**:
We need to find the volume of the space that needs to be filled, which is the volume of the cube minus the volume of the baseball:
\[
\text{Volume of acrylic} = \text{Volume of cube} - \text{Volume of sphere} = 27 \, \text{cubic inches} - 14.137 \, \text{cubic inches} = 12.863 \, \text{cubic inches}
\]
4. **Convert the volume from cubic inches to fluid ounces**:
Given that \( 1 \, \text{cubic inch} = 0.554 \, \text{fluid ounces} \):
\[
\text{Volume in fluid ounces} = 12.863 \, \text{cubic inches} \times 0.554 \, \text{fluid ounces/cubic inch} \approx 7.13 \, \text{fl oz}
\]
Thus, the amount of acrylic needed to fill the remaining space in the container is approximately **7.13 fluid ounces**. The answer is:
**7.13 fl oz**
1. **Calculate the volume of the cube**:
\[
\text{Volume of cube} = \text{side}^3 = 3 \, \text{in} \times 3 \, \text{in} \times 3 \, \text{in} = 27 \, \text{cubic inches}
\]
2. **Calculate the volume of the baseball**:
The baseball is a sphere that perfectly inscribes within the cube. The diameter of the baseball is equal to the side length of the cube, which is 3 inches. Therefore, the radius \( r \) of the baseball is:
\[
r = \frac{\text{diameter}}{2} = \frac{3 \, \text{in}}{2} = 1.5 \, \text{in}
\]
The volume of a sphere is calculated using the formula:
\[
\text{Volume of sphere} = \frac{4}{3} \pi r^3
\]
Substituting the value of \( r \):
\[
\text{Volume of sphere} = \frac{4}{3} \pi (1.5)^3 = \frac{4}{3} \pi (3.375) = \frac{4 \cdot 3.375 \pi}{3} = 4.5 \pi \approx 14.137 \, \text{cubic inches}
\]
3. **Calculate the volume of the acrylic epoxy**:
We need to find the volume of the space that needs to be filled, which is the volume of the cube minus the volume of the baseball:
\[
\text{Volume of acrylic} = \text{Volume of cube} - \text{Volume of sphere} = 27 \, \text{cubic inches} - 14.137 \, \text{cubic inches} = 12.863 \, \text{cubic inches}
\]
4. **Convert the volume from cubic inches to fluid ounces**:
Given that \( 1 \, \text{cubic inch} = 0.554 \, \text{fluid ounces} \):
\[
\text{Volume in fluid ounces} = 12.863 \, \text{cubic inches} \times 0.554 \, \text{fluid ounces/cubic inch} \approx 7.13 \, \text{fl oz}
\]
Thus, the amount of acrylic needed to fill the remaining space in the container is approximately **7.13 fluid ounces**. The answer is:
**7.13 fl oz**