\(\frac{7}{3}\) cm
Calculate Sphere Volume and Cylinder Water Rise
The problem asks for the increase in water level within a cylinder when a sphere is fully submerged. This water level rise is equivalent to the volume of the sphere divided by the cross-sectional area of the cylinder.
Sphere Dimensions and Volume Calculation
The diameter of the sphere is given as 14 cm.
- Sphere radius (\(r\)): \(r = \frac{\text{diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm}\)
The formula for the volume of a sphere is:
\(V_{\text{sphere}} = \frac{4}{3} \pi r^3\)
Substitute the sphere's radius into the formula:
\(V_{\text{sphere}} = \frac{4}{3} \pi (7 \text{ cm})^3 = \frac{4}{3} \pi (343) \text{ cm}^3\)
Cylinder Dimensions and Water Rise Volume
The radius of the cylindrical vessel (\(R\)) is given as 14 cm.
When the sphere is completely submerged, the volume of water displaced equals the sphere's volume. This displaced volume causes the water level in the cylinder to rise.
The volume of the water rise (\(V_{\text{rise}}\)) forms a cylinder with radius \(R\) and height \(h\) (the water level rise).
The formula for the volume of the water rise is:
\(V_{\text{rise}} = \pi R^2 h\)
Substitute the cylinder's radius:
\(V_{\text{rise}} = \pi (14 \text{ cm})^2 h = \pi (196) h \text{ cm}^3\)
Equating Volumes and Solving for Height
The volume of the submerged sphere must equal the volume of the water rise in the cylinder:
\(V_{\text{sphere}} = V_{\text{rise}}\)
\(\frac{4}{3} \pi (343) \text{ cm}^3 = \pi (196) h \text{ cm}^3\)
To find the height (\(h\)), we can cancel \(\pi\) from both sides and rearrange the equation:
\(\frac{4}{3} \times 343 = 196 h\)
\(h = \frac{4 \times 343}{3 \times 196}\)
Simplify the expression:
- \(343 = 7^3\)
- \(196 = 4 \times 49 = 4 \times 7^2\)
- \(h = \frac{4 \times 7^3}{3 \times (4 \times 7^2)}\)
- Cancel common terms (4 and \(7^2\)):
- \(h = \frac{7}{3} \text{ cm}\)
Final Answer
The water level rise is calculated to be \(\frac{7}{3}\) cm.
Similar Questions
- Let x be the length of a diagonal of a face of a cube and y be the length of a diagonal of the cube. If x + y = \((5 + 2\sqrt{6})\) units, then what is the total surface area of the cube ?
- The diameter of a copper solid sphere is 6 cm. The sphere is melted and recast into a wire. If the diameter of the wire is 0.5 cm, then what is the length of the wire ?
- A metallic sheet of rectangular shape is 48 cm × 36 cm. From each corner a square of side x cm is cut off and an open box is made of the remaining sheet. If the box holds 5.12 litre of water, then what is the value of x ?
Important Questions from 3-D Mensuration
Find the total surface area of a closed cylinder having a base radius of 70 m and a height of 110 m. [Use π = \(22\over7\)]
Which of the following is a geometrical figure with a three-dimensional geometry that has eight vertices and six rectangular faces?
- The curved surface area of a cylinder is half of its total surface area. If its height is 195 cm, then find its diameter (in cm).
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone.
(Use $\pi = \frac{22}{7}$)- A cylindrical rod has an curved surface area of $4,900 \text{ cm}^2$. If the length of the rod is 97 cm, then the radius (in cm) of the rod, correct to two places of decimal, is:
Take $\pi = \frac{22}{7}$
