A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter o the spherical part is 8.5 cm. By measuring the amount of water it holds, a child finds its volume to be 345 cm3. Check whether she is correct, taking the above as the inside measurements, and \(\pi = 3.14.\)
Solution and Explanation
Height (h) of cylindrical part = 8 cm
Radius (r2) of cylindrical part = 1 cm
Radius (r1) spherical part = 4.25 cm
Volume of vessel = Volume of sphere + Volume of cylinder
\(=\frac{4}{3}\pi r^3_1+\pi r_2^2h\)
\(=\frac{4}{3}\pi(8.52)^3+\pi(1)^2(8)\)
\(=\frac{4}{3}×3.14×76.765625+8×3.14\)
\(=321.392+25.12\)
\(=346.51\ \text{cm}^3\)
Hence, she is wrong.
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Concepts Used:
Surface Area and Volume
Surface area and volume are two important concepts in geometry that are used to measure the size and shape of three-dimensional objects.
Surface area is the measure of the total area that the surface of an object covers. It is expressed in square units, such as square meters or square inches. To calculate the surface area of an object, we find the area of each face or surface and add them together. For example, the surface area of a cube is equal to six times the area of one of its faces.
Volume, on the other hand, is the measure of the amount of space that an object takes up. It is expressed in cubic units, such as cubic meters or cubic feet. To calculate the volume of an object, we measure the length, width, and height of the object and multiply these three dimensions together. For example, the volume of a cube is equal to the length of one of its edges cubed.
Surface area and volume are important in many fields, such as architecture, engineering, and manufacturing. For example, surface area is used to calculate the amount of material needed to cover an object, while volume is used to determine the amount of space that a container can hold. Understanding surface area and volume is also important in calculus and physics, where they are used to model the behavior of objects in three-dimensional space.