Answer:
See attachment.
Step-by-step explanation:
Diameter and Radius
The diameter of a circle is twice its radius. Therefore, to find the radius from the diameter, divide the diameter by 2. Similarly, to find the diameter from the radius, multiply the radius by 2.
[tex]\dotfill[/tex]
Base Area
The base area of both a cylinder and a cone is given by πr², where 'r' represents the radius. Therefore, to find the base area, square the radius and multiply it by π.
[tex]\dotfill[/tex]
Cylinder Volume
The formula for the volume of a cylinder is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Cylinder}}\\\\V=\pi r^2 h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
Given that the base area of a cylinder is πr², its volume is essentially its base area multiplied by its height. Therefore, to find the volume of the cylinder, simply multiply the base area by the given height.
[tex]\dotfill[/tex]
Cone Volume
The formula for the volume of a cone is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Volume of a Cone}}\\\\V=\dfrac{1}{3}\pi r^2h\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$V$ is the volume.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}[/tex]
Comparing the formula for the volume of a cone to the formula for the volume of a cylinder, we can see that the volume of a cone is one-third the volume of a cylinder, provided both figures have the same radius and height.
Therefore, to find the volume of a cylinder, multiply the volume of the cone by 3. Similarly, to find the volume of the cone, divide the volume of the cylinder by 3.
[tex]\dotfill[/tex]
Row 1
Given that the radius (r) is 5 units, then:
[tex]\textsf{Diameter}= 2r=2 \cdot 5 = 10[/tex]
[tex]\textsf{Base Area} = \pi r^2 = \pi \cdot 5^2 = 25\pi[/tex]
Given that the height (h) is 7 units, then:
[tex]\textsf{Cylinder Volume}=\textsf{Base Area} \cdot h=25\pi \cdot 7=175\pi[/tex]
[tex]\textsf{Cone Volume}=\dfrac{\textsf{Cylinder Volume}}{3}=\dfrac{175}{3}\pi[/tex]
[tex]\dotfill[/tex]
Row 2
Given that the diameter is 6 units, then:
[tex]\textsf{Radius $(r)$}=\dfrac{\textsf{Diameter}}{2}=\dfrac{6}{2}=3[/tex]
[tex]\textsf{Base Area} = \pi r^2 = \pi \cdot 3^2 = 9\pi[/tex]
Given that the volume of the cone is 40π, then:
[tex]\textsf{Cylinder Volume}=\textsf{Cone Volume} \cdot 3=40\pi \cdot 3=120\pi[/tex]
[tex]\textsf{Height}=\dfrac{\textsf{Cylinder Volume}}{\textsf{Base Area}}=\dfrac{120\pi}{9\pi}=\dfrac{40}{3}[/tex]
[tex]\dotfill[/tex]
Row 3
Given that the Base Area is 36π, then:
[tex]\textsf{Radius $(r)$}=\sqrt{\dfrac{\textsf{Base Area}}{\pi}}=\sqrt{\dfrac{36\pi}{\pi}}=\sqrt{36}=6[/tex]
[tex]\textsf{Diameter}= 2r=2 \cdot 6 = 12[/tex]
Given that the volume of the cone is 48π, then:
[tex]\textsf{Cylinder Volume}=\textsf{Cone Volume} \cdot 3=48\pi \cdot 3=144\pi[/tex]
[tex]\textsf{Height}=\dfrac{\textsf{Cylinder Volume}}{\textsf{Base Area}}=\dfrac{144\pi}{36\pi}=4[/tex]
