Final answer:
The volume of the cone is found by first determining the radius from the circumference, then calculating the height, and finally applying the volume formula. The correct volume calculation does not match any of the provided answer choices (A, B, C, or D).
Step-by-step explanation:
The question involves finding the volume of a cone given the base circumference and a relationship between the height and the radius of the cone. The formula to calculate the volume (V) of a cone is V = (1/3)πr^2h, where r is the radius and h is the height. To begin, we need to determine the radius (r) from the given circumference, which is C = 2πr. Then, we can find the height (h) using the provided relationship that it is 6 centimeters less than twice the radius (h = 2r - 6). Using these values, we can compute the volume in terms of π and also provide a rounded decimal value.
Step-by-Step Calculation
1. Find the radius by using the circumference formula C = 2πr:
19π cm = 2πr
r = 19π cm / 2π = 9.5 cm.
2. Determine the height with the given relationship:
h = 2r - 6
h = 2(9.5 cm) - 6 cm
h = 19 cm - 6 cm
h = 13 cm.
3. Calculate the volume:
V = (1/3)πr^2h
V = (1/3)π(9.5 cm)^2(13 cm)
V = (1/3)π90.25 cm^2(13 cm)
V = (1/3)π1173.25 cm^3
V = 391.083333π cm^3
V ≈ 1227.4 cm^3 (rounded to the nearest tenth).
Based on our calculations, none of the provided answer choices (A, B, C, or D) match our calculated volume.