To determine the radius of the spherical object that was immersed in the water, we can use the concept of volume displacement. The volume of water displaced by the spherical object will equal the volume of the spherical object itself.
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Calculate the initial volume of water in the measuring cylinder:
- The formula for the volume \( V \) of a cylinder is given by: \[ V = \pi r^2 h \] Where \( r \) is the radius and \( h \) is the height of the water.
Given:
- Base radius \( r = 5 \) cm
- Height of water \( h_1 = 6 \) cm
Thus, the initial volume of water is: \[ V_{initial} = \pi (5)^2 (6) = \pi (25)(6) = 150\pi , \text{cm}^3 \]
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Calculate the volume of water after the spherical object is immersed:
- The new height of water \( h_2 = 10 \) cm
The new volume of water is: \[ V_{final} = \pi (5)^2 (10) = \pi (25)(10) = 250\pi , \text{cm}^3 \]
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Calculate the volume of the spherical object:
- The volume of the displaced water (which is equal to the volume of the spherical object) can be found by subtracting the initial volume from the final volume: \[ V_{displaced} = V_{final} - V_{initial} = 250\pi - 150\pi = 100\pi , \text{cm}^3 \]
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Determine the radius of the spherical object:
- The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] Setting this equal to the volume of the spherical object: \[ \frac{4}{3} \pi r^3 = 100\pi \]
Divide both sides by \( \pi \): \[ \frac{4}{3} r^3 = 100 \]
Multiply both sides by \( \frac{3}{4} \): \[ r^3 = 100 \times \frac{3}{4} = 75 \]
Now, take the cube root of both sides to find \( r \): \[ r = \sqrt[3]{75} \]
Calculating \( r \): \[ r \approx 4.22 , \text{cm} \]
Thus, the radius of the spherical object is approximately \( 4.22 , \text{cm} \).