Answer:
Option B 1 : 3
Step-by-step explanation:
Volume formulas & definitions:
To solve this problem, we must remember the formulas for volume of each shape:
[tex]V_{cylinder}=\pi r^2h\\V_{cone}=\frac{1}{3} \pi r^2 h[/tex] where r represents the radius of the circular part of the shape, and h represents the height of the shape (the plural of height is "heights", and the plural of radius is "radii")
So, when the question states that the shapes "have congruent heights and radii" it means that the height of both objects is the same, and the radius of both objects is the same.
Ratios
To find the ratio of the volume of the cone to the volume of the cylinder, we must setup the ratio. Ratios can be setup as a fraction where the first quantity is the numerator (top of fraction), and the second quantity is the denominator (bottom of the fraction):
[tex]\dfrac{V_{cone}}{V_{cylinder}}=\dfrac{\frac{1}{3} \pi r^2 h} {\pi r^2 h}[/tex]
Since the pi, the "r", and the "h" are the same in both the numerator and denominator (and since there is only multiplication & division in the fraction) these common factors can cancel, and reduce the fraction to the following:
[tex]\dfrac{V_{cone}}{V_{cylinder}}=\dfrac{\frac{1}{3}} {1}[/tex]
Any number divided by 1 doesn't change, so one-third divided by 1 is still one-third.
[tex]\dfrac{V_{cone}}{V_{cylinder}}=\frac{1}{3}[/tex]
Other representations of ratios
Lastly, we must remember that a ratio an also be written with a colon symbol, where the numerator is written first, and the denominator is written second.
So, the ratio of the volume of the cone to the volume of the cylinder is:
1 : 3
