To solve this problem step-by-step, we'll break down the given scenario involving the calculation of the volumes for two geometric shapes: a rectangular prism and a cylinder.
Firstly, we have the following measurements for the rectangular prism:
- Length ([tex]\( l \)[/tex]): 5 units
- Width ([tex]\( w \)[/tex]): 7 units
- Height ([tex]\( h \)[/tex]): 3 units
To find the volume [tex]\( V \)[/tex] of a rectangular prism, the formula is:
[tex]\[ V = l \times w \times h \][/tex]
Plugging in the values:
[tex]\[ V = 5 \times 7 \times 3 \][/tex]
[tex]\[ V = 35 \times 3 \][/tex]
[tex]\[ V = 105 \][/tex]
So, the volume of the rectangular prism is [tex]\( 105 \)[/tex] cubic units.
Next, we move on to the cylinder, which has the following measurements:
- Radius ([tex]\( r \)[/tex]): 4 units
- Height ([tex]\( h \)[/tex]): 6 units
The formula for the volume [tex]\( V \)[/tex] of a cylinder is:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( \pi \)[/tex] (pi) is approximately 3.141592653589793.
Plugging in the values:
[tex]\[ V = \pi \times 4^2 \times 6 \][/tex]
[tex]\[ V = \pi \times 16 \times 6 \][/tex]
[tex]\[ V = \pi \times 96 \][/tex]
When you multiply 96 by the value of [tex]\( \pi \)[/tex]:
[tex]\[ V \approx 3.141592653589793 \times 96 \][/tex]
[tex]\[ V \approx 301.59289474462014 \][/tex]
Thus, the volume of the cylinder is approximately [tex]\( 301.59 \)[/tex] cubic units.
So, to summarize:
- The volume of the rectangular prism is [tex]\( 105 \)[/tex] cubic units.
- The volume of the cylinder is approximately [tex]\( 301.59 \)[/tex] cubic units.
These detailed calculations confirm the volumes of the rectangular prism and the cylinder based on their given dimensions.
