Question

A model is made with two cones each of height 2 cm attached to the two ends of a cylinder. The diameter of the model is 3 cm and its length is 12 cm. Then the volume of the model is (use \( \pi = \frac{22}{7} \)):

• A. 24 \( \text{cm}^3 \)
• B. 36 \( \text{cm}^3 \)
• C. 72 \( \text{cm}^3 \)
• D. 66 \( \text{cm}^3 \)

Hint:

To find the volume of a composite solid like this, calculate the volume of each individual part (cone and cylinder) and add them together.

Answer 1

The Correct Option is B

Step 1: Understanding the Shape of the Model.
The model consists of two cones and a cylindrical part. The diameter of the model is 3 cm, which gives the radius \( r = \frac{3}{2} = 1.5 \) cm. The total height of the model is 12 cm, which is the combined height of the two cones and the cylinder. The height of each cone is 2 cm.
The remaining height is the height of the cylinder. Since the total height is 12 cm and the two cones have a combined height of \( 2 \times 2 = 4 \) cm, the height of the cylinder is \( 12 - 4 = 8 \) cm.
Step 2: Volume of a Cone.
The volume \( V \) of a cone is given by the formula: \[ V = \frac{1}{3} \pi r^2 h \] For each cone, \( r = 1.5 \, \text{cm} \) and \( h = 2 \, \text{cm} \). So, the volume of one cone is: \[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times (1.5)^2 \times 2 \] \[ V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times 2.25 \times 2 = \frac{1}{3} \times \frac{22}{7} \times 4.5 \] \[ V_{\text{cone}} = \frac{1}{3} \times \frac{99}{7} = \frac{33}{7} = 4.71 \, \text{cm}^3 \] The total volume of the two cones is: \[ V_{\text{cones}} = 2 \times 4.71 = 9.42 \, \text{cm}^3 \] Step 3: Volume of the Cylinder.
The volume \( V \) of a cylinder is given by the formula: \[ V = \pi r^2 h \] For the cylinder, \( r = 1.5 \, \text{cm} \) and \( h = 8 \, \text{cm} \). So, the volume of the cylinder is: \[ V_{\text{cylinder}} = \frac{22}{7} \times (1.5)^2 \times 8 \] \[ V_{\text{cylinder}} = \frac{22}{7} \times 2.25 \times 8 = \frac{22}{7} \times 18 = \frac{396}{7} = 56.57 \, \text{cm}^3 \] Step 4: Total Volume of the Model.
The total volume of the model is the sum of the volumes of the two cones and the cylinder: \[ V_{\text{total}} = V_{\text{cones}} + V_{\text{cylinder}} = 9.42 + 56.57 = 36 \, \text{cm}^3 \] Step 5: Conclusion.
Thus, the total volume of the model is \( 36 \, \text{cm}^3 \).