To solve this problem, let's begin by calculating the volume of the original cylinder with a radius of 4 inches and a height of 5 inches.
The formula for the volume of a cylinder is:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
1. Calculate the volume of the original cylinder:
- Radius ([tex]\( r \)[/tex]) = 4 inches
- Height ([tex]\( h \)[/tex]) = 5 inches
[tex]\[
V_{\text{original}} = \pi (4^2) (5) = \pi (16) (5) = 80\pi
\][/tex]
[tex]\[
V_{\text{original}} \approx 251.327 \, \text{cubic inches} \quad (\text{using } \pi \approx 3.14159)
\][/tex]
2. Check the new set of dimensions:
First set of new dimensions:
- Radius ([tex]\( r \)[/tex]) = 8 inches
- Height ([tex]\( h \)[/tex]) = 5 inches
[tex]\[
V_{\text{new1}} = \pi (8^2) (5) = \pi (64) (5) = 320\pi
\][/tex]
[tex]\[
V_{\text{new1}} \approx 1005.312 \, \text{cubic inches}
\][/tex]
Comparison:
[tex]\[
V_{\text{new1}} \neq V_{\text{original}}
\][/tex]
The volume with these dimensions is not the same as the original.
Second set of new dimensions:
- Radius ([tex]\( r \)[/tex]) = 2 inches
- Height ([tex]\( h \)[/tex]) = 20 inches
[tex]\[
V_{\text{new2}} = \pi (2^2) (20) = \pi (4) (20) = 80\pi
\][/tex]
[tex]\[
V_{\text{new2}} \approx 251.327 \, \text{cubic inches}
\][/tex]
Comparison:
[tex]\[
V_{\text{new2}} = V_{\text{original}}
\][/tex]
The volume with these dimensions is the same as the original.
In conclusion:
- The volume of a cylinder with radius 8 inches and height 5 inches is different from the original cylinder's volume.
- The volume of a cylinder with radius 2 inches and height 20 inches is the same as the original cylinder's volume.
