To determine which statement is true regarding the volumes of the vases, we can use the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
Now we will calculate the volumes of both vases, A and B.
Volume of Vase A:
- Height (h) = 14 inches
- Radius (r) = 2 inches
\[ V_A = \pi r^2 h = \pi (2^2)(14) = \pi (4)(14) = 56\pi \]
Volume of Vase B:
- Height (h) = 7 inches
- Radius (r) = 4 inches
\[ V_B = \pi r^2 h = \pi (4^2)(7) = \pi (16)(7) = 112\pi \]
Now we have the volumes:
- Volume of Vase A, \( V_A = 56\pi \)
- Volume of Vase B, \( V_B = 112\pi \)
Now, let's check each of the statements:
A. The volume of vase B is twice the volume of vase A.
\[ 2 V_A = 2(56\pi) = 112\pi = V_B \quad \text{(True)} \]
B. The volume of vase A is twice the volume of vase B.
\[ 2 V_B = 2(112\pi) = 224\pi \quad \text{(False)} \]
C. The volumes of the two vases are equal.
\[ V_A \neq V_B \quad \text{(False)} \]
D. The volume of vase B is four times the volume of vase A.
\[ 4 V_A = 4(56\pi) = 224\pi \quad \text{(False)} \]
The correct statement is:
A. The volume of vase B is twice the volume of vase A.