Question 13 of 13intermediate⚖️ EvaluateNumerical4 marks

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.

Correct Answer

To check the correctness of the child's measurement, we calculate the actual volume of the vessel.

Step 1: Find the dimensions. The cylindrical neck has a diameter of 2 cm, so its radius (r) is 2/2=1 cm2/2 = 1 \text{ cm}. The height (h) of the neck is 8 cm. The spherical part has a diameter of 8.5 cm, so its radius (R) is 8.5/2=4.25 cm8.5/2 = 4.25 \text{ cm}.

Step 2: Calculate the volume of the cylindrical neck. Using the formula for the volume of a cylinder, V=πr2hV = \pi r^2 h: Volume of cylinder =3.14×(1)2×8= 3.14 \times (1)^2 \times 8 =3.14×1×8=25.12 cm3= 3.14 \times 1 \times 8 = 25.12 \text{ cm}^3.

Step 3: Calculate the volume of the spherical part. From the context, the volume of a hemisphere is 23πr3\frac{2}{3} \pi r^3. Since a sphere consists of two hemispheres, its volume is 43πR3\frac{4}{3} \pi R^3. Volume of sphere =43×3.14×(4.25)3= \frac{4}{3} \times 3.14 \times (4.25)^3

=43×3.14×76.7656= \frac{4}{3} \times 3.14 \times 76.7656

=321.39 cm3= 321.39 \text{ cm}^3 (approx).

Step 4: Calculate the total volume and compare. Total Volume == Volume of cylinder ++ Volume of sphere

=25.12+321.39= 25.12 + 321.39

=346.51 cm3= 346.51 \text{ cm}^3.

The child found the volume to be 345 cm3345 \text{ cm}^3. Since 346.51 cm3346.51 \text{ cm}^3 is not equal to 345 cm3345 \text{ cm}^3, she is not correct.

Exercise: EXERCISE 12.2 | Q: 8 | (Page Page 10)
For More Understanding

Explanation

The solution follows the methodology shown in Example 6 of the provided context, which calculates the volume of a composite solid (a cylinder and a hemisphere). Here, we calculate the volume of the cylindrical neck and the spherical body separately. The formula for the volume of a sphere (43πr3\frac{4}{3}\pi r^3) is derived from the hemisphere formula (23πr3\frac{2}{3}\pi r^3) given in the context. The sum of these volumes gives the actual capacity, which is then compared to the child's measurement to verify correctness.

Solution Steps

  1. Step 1: Determine the radius of the cylindrical neck (1 cm) and the spherical part (4.25 cm).

  2. Step 2: Calculate the volume of the cylindrical neck using V=πr2hV = \pi r^2 h, resulting in 25.12 cm325.12 \text{ cm}^3.

  3. Step 3: Calculate the volume of the spherical part using V=43πR3V = \frac{4}{3} \pi R^3, resulting in approximately 321.39 cm3321.39 \text{ cm}^3.

  4. Step 4: Add the two volumes to get the total volume (346.51 cm3346.51 \text{ cm}^3) and conclude that the child's measurement of 345 cm3345 \text{ cm}^3 is incorrect.

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