Answer :
To solve the problem of calculating how many cubic inches of plastic it takes to make the plastic dog dish, we need to consider both the cylindrical and spherical parts of the dish.
The dish consists of two parts:
1. A cylinder which forms the main body.
2. A half-sphere (hemisphere) which forms the bowl.
1. Cylinder:
- Radius: The radius of the cylinder includes the bowl radius plus the width of the rim. Assuming the bowl radius is 2 inches and the rim is 0.5 inches wide, the cylinder's radius is [tex]\(2.5\)[/tex] inches.
- Height: Since the bowl is shaped like a half-sphere, the height of the cylinder is equivalent to the radius of the sphere (diameter is 2 inches, so the radius is 1 inch).
- Volume of the Cylinder: The volume [tex]\(V\)[/tex] of a cylinder is calculated using the formula:
[tex]\[
V = \pi \times \text{radius}^2 \times \text{height}
\][/tex]
Using the given dimensions,
[tex]\[
V = \pi \times (2.5)^2 \times 1 = 19.63\, \text{cubic inches (approximately)}
\][/tex]
2. Hemisphere:
- Radius: The radius of the sphere is 2 inches.
- Volume of the Hemisphere: The volume [tex]\(V\)[/tex] of a hemisphere is half of the sphere's volume. The sphere's volume formula is:
[tex]\[
V = \frac{4}{3} \pi \text{radius}^3
\][/tex]
So, for the hemisphere:
[tex]\[
V = \frac{1}{2} \times \frac{4}{3} \pi \times (2)^3 = 16.76\, \text{cubic inches (approximately)}
\][/tex]
Total Volume:
Add the volume of the cylindrical part to the volume of the hemispherical part to get the total volume of the plastic used:
- Total Volume = Volume of Cylinder + Volume of Hemisphere
[tex]\[
\text{Total Volume} = 19.63 + 16.76 = 36.39\, \text{cubic inches (approximately)}
\][/tex]
Rounding this total to the nearest cubic inch gives approximately 36 cubic inches.
In the multiple choice provided, the closest value to our calculation is not one of the options given. Therefore, using the information from the numerical result mentioned before, the likely intended closest answer or correction should be checked, and perhaps the correct calculation should have resulted in answer A: 41. However, we're certain that the correct calculated cubic inches for the current problem structure resulted in approximately 36.
The dish consists of two parts:
1. A cylinder which forms the main body.
2. A half-sphere (hemisphere) which forms the bowl.
1. Cylinder:
- Radius: The radius of the cylinder includes the bowl radius plus the width of the rim. Assuming the bowl radius is 2 inches and the rim is 0.5 inches wide, the cylinder's radius is [tex]\(2.5\)[/tex] inches.
- Height: Since the bowl is shaped like a half-sphere, the height of the cylinder is equivalent to the radius of the sphere (diameter is 2 inches, so the radius is 1 inch).
- Volume of the Cylinder: The volume [tex]\(V\)[/tex] of a cylinder is calculated using the formula:
[tex]\[
V = \pi \times \text{radius}^2 \times \text{height}
\][/tex]
Using the given dimensions,
[tex]\[
V = \pi \times (2.5)^2 \times 1 = 19.63\, \text{cubic inches (approximately)}
\][/tex]
2. Hemisphere:
- Radius: The radius of the sphere is 2 inches.
- Volume of the Hemisphere: The volume [tex]\(V\)[/tex] of a hemisphere is half of the sphere's volume. The sphere's volume formula is:
[tex]\[
V = \frac{4}{3} \pi \text{radius}^3
\][/tex]
So, for the hemisphere:
[tex]\[
V = \frac{1}{2} \times \frac{4}{3} \pi \times (2)^3 = 16.76\, \text{cubic inches (approximately)}
\][/tex]
Total Volume:
Add the volume of the cylindrical part to the volume of the hemispherical part to get the total volume of the plastic used:
- Total Volume = Volume of Cylinder + Volume of Hemisphere
[tex]\[
\text{Total Volume} = 19.63 + 16.76 = 36.39\, \text{cubic inches (approximately)}
\][/tex]
Rounding this total to the nearest cubic inch gives approximately 36 cubic inches.
In the multiple choice provided, the closest value to our calculation is not one of the options given. Therefore, using the information from the numerical result mentioned before, the likely intended closest answer or correction should be checked, and perhaps the correct calculation should have resulted in answer A: 41. However, we're certain that the correct calculated cubic inches for the current problem structure resulted in approximately 36.
