A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm

Updated by Tiwari Academy
on November 28, 2023, 10:32 AM

To check the child’s measurement of the volume of the spherical glass vessel, we calculate the volume of both the spherical part and the cylindrical neck and sum them.
The diameter of the spherical part is 8.5 cm, giving a radius of 4.25 cm. The volume of a sphere is (4/3)πr³. So, the volume of the spherical part is (4/3)×3.14×4.25³.
The cylindrical neck is 8 cm long with a diameter of 2 cm, giving a radius of 1 cm. The volume of a cylinder is πr²h. So, the volume of the neck is 3.14×1²×8.
Adding the volumes of the spherical part and the cylindrical neck gives the total volume of the vessel. This total volume can be compared with the child’s measurement of 345 cm³ to check its accuracy.

A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of 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.

Let’s discuss in detail

Volume Calculation of a Composite Object

The task involves verifying the accuracy of a child’s measurement of the volume of a spherical glass vessel with a cylindrical neck. This problem is a practical application of geometry, specifically in calculating the volume of composite shapes. The vessel consists of two distinct parts: a spherical section and a cylindrical neck. The spherical part has a diameter of 8.5 cm, and the cylindrical neck is 8 cm long with a diameter of 2 cm. Using these dimensions, we aim to calculate the total volume of the vessel and compare it with the child’s measurement of 345 cm³ to determine its accuracy.

Calculating the Volume of the Spherical Part

The first step is to calculate the volume of the spherical part of the vessel. The formula for the volume of a sphere is (4/3)πr³, where r is the radius. The diameter of the sphere is given as 8.5 cm, which makes the radius 4.25 cm (half of the diameter). Substituting these values into the formula, we get (4/3)×3.14×4.25³ cubic centimeters as the volume of the spherical part. This calculation is crucial as it represents the major portion of the vessel’s volume.

Determining the Volume of the Cylindrical Neck

Next, we calculate the volume of the cylindrical neck of the vessel. The formula for the volume of a cylinder is πr²h, where r is the radius and h is the height (or length). The neck has a diameter of 2 cm, giving a radius of 1 cm, and a length of 8 cm. Therefore, the volume of the cylindrical neck is 3.14×1²×8 cubic centimeters. This volume, though smaller than that of the spherical part, is significant in contributing to the total volume of the vessel.

Total Volume of the Glass Vessel
To find the total volume of the glass vessel, we add the volumes of the spherical part and the cylindrical neck. The sum of these volumes gives the total volume of the vessel. This step is essential because it combines the volumes of the two different geometric shapes that make up the vessel, providing a comprehensive calculation of its capacity.

Comparing Calculated Volume with the Child’s Measurement

With the total volume calculated, the next step is to compare it with the child’s measurement of 345 cm³. This comparison is crucial to determine the accuracy of the child’s measurement. If the calculated volume closely matches the child’s measurement, it validates her findings. If there is a significant discrepancy, it may indicate a miscalculation or an error in the measurement process.

Importance of Accurate Volume Calculation

In conclusion, this exercise demonstrates the importance of accurate volume calculation in everyday scenarios. It highlights the practical application of geometric principles in real-life situations, such as measuring the capacity of containers. This problem-solving approach is not just about applying mathematical formulas; it’s about understanding the spatial properties of objects and their practical implications. Such calculations are essential in various fields, including manufacturing, cooking, and science experiments, where precise measurements are crucial for success and accuracy.

Discuss this question in detail or visit to Class 10 Maths Chapter 12 for all questions.
Questions of 10th Maths Exercise 12.2 in Detail

A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of π.

Rachel, an engineering student, was asked to make a model shaped like a cylinder with two cones attached at its two ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model that Rachel made.

A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm.

A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens. The dimensions of the cuboid are 15 cm by 10 cm by 3.5 cm. The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm. Find the volume of wood in the entire stand.

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm³ of iron has approximately 8g mass.

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

A spherical glass vessel has a cylindrical neck 8 cm long, 2 cm in diameter; the diameter of 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 π = 3.14.