To find how much water is in the cylinder after placing the 6-inch-diameter ball inside it, we first need to calculate the volume of the cylinder and then subtract the volume of the ball.
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Calculate the volume of the cylinder. The formula for the volume of a cylinder is:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.
The diameter of the cylinder is 10 inches, so the radius \( r \) is:
\[ r = \frac{10}{2} = 5 \text{ inches} \]
The height \( h \) of the cylinder is 15 inches. Now, we can calculate the volume:
\[ V_{\text{cylinder}} = \pi (5)^2 (15) = \pi (25)(15) = 375\pi \text{ cubic inches} \]
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Calculate the volume of the ball. The formula for the volume of a sphere is:
\[ V = \frac{4}{3} \pi r^3 \]
The diameter of the ball is 6 inches, so the radius \( r \) is:
\[ r = \frac{6}{2} = 3 \text{ inches} \]
Now we can calculate the volume of the ball:
\[ V_{\text{ball}} = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi \text{ cubic inches} \]
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Calculate the volume of water in the cylinder. The volume of water is the volume of the cylinder minus the volume of the ball:
\[ V_{\text{water}} = V_{\text{cylinder}} - V_{\text{ball}} = 375\pi - 36\pi = 339\pi \text{ cubic inches} \]
Thus, the amount of water in the cylinder is:
\[ \boxed{339\pi} \text{ cubic inches} \]