To solve the problem of finding the radius of the cylinder when a spherical metal bob is immersed in water, we need to follow a systematic approach. Let's break it down step-by-step. ### Step 1: Understand the Problem We have a spherical metal bob with a radius of 3 cm that is fully immersed in a cylinder of water. When the bob is submerged, the water level in the cylinder rises by 1 cm. Our goal is to find the radius of the cylinder. ### Step 2: Calculate the Volume of the Spherical Bob The volume \( V \) of a sphere can be calculated using the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. For our spherical bob: - Radius \( r = 3 \) cm Substituting the value into the formula: \[ V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi \text{ cm}^3 \] ### Step 3: Relate the Volume of the Bob to the Rise in Water Level When the bob is submerged, it displaces an amount of water equal to its own volume. This displaced volume causes the water level in the cylinder to rise. Let’s denote: - \( h \) = rise in water level = 1 cm - \( R \) = radius of the cylinder (unknown) The volume of water displaced, which is equal to the volume of the bob, can also be expressed in terms of the cylinder's dimensions: \[ \text{Volume of displaced water} = \text{Base Area of Cylinder} \times \text{Height of Water Rise} \] The base area \( A \) of the cylinder is given by: \[ A = \pi R^2 \] Thus, the volume of the displaced water is: \[ \text{Volume} = \pi R^2 \times h = \pi R^2 \times 1 = \pi R^2 \text{ cm}^3 \] ### Step 4: Set the Two Volumes Equal Since the volume of the bob equals the volume of the displaced water: \[ 36\pi = \pi R^2 \] ### Step 5: Solve for the Radius of the Cylinder We can simplify the equation by dividing both sides by \( \pi \): \[ 36 = R^2 \] Now, taking the square root of both sides: \[ R = \sqrt{36} = 6 \text{ cm} \] ### Conclusion The radius of the cylinder is **6 cm**. Therefore, the correct option is **D**. ### Explanation of Other Options - **A. 12 cm**: This option is incorrect because it suggests a larger radius than necessary to account for the volume displaced by the bob. - **B. 1 cm**: This option is incorrect as it is far too small to accommodate the volume of the bob, which would not cause a 1 cm rise in water level. - **C. 3 cm**: This option is also incorrect because it does not provide enough area to displace the volume of the bob, leading to a smaller rise than observed. ### Revision Summary - The volume of a sphere is calculated using \( V = \frac{4}{3} \pi r^3 \). - The volume of water displaced equals the volume of the bob when submerged. - The relationship between the rise in water level and the cylinder's radius is given by \( \pi R^2 \times h \). - The final calculation shows that the radius of the cylinder is 6 cm.