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A round pencil is sharpened to a cone shape at both ends as shown in the image below. Use the data in the image and the value 22/7 for π to find the volume of the pencil.

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Question: 

A round pencil is sharpened to a cone shape at both ends as shown in the image below. Use the data in the image and the value 22/7 for π to find the volume of the pencil.

 

Solution

To find the volume of the pencil, we need to calculate the volume of two parts:

  1. The cylindrical part (the main body of the pencil)
  2. The two conical tips (sharpened ends)

Given dimensions:

  • Cylinder (main body of pencil)

    • Radius = 12=0.5\frac{1}{2} = 0.5
    • Height = 16 cm

  • Each cone (sharpened tip)

    • Radius = 0.50.5cm (same as the cylinder)
    • Height = 1.21.2 cm
    • There are two cones

Step 1: Volume of the cylindrical part

The volume of a cylinder is given by:

V=πr2hV = \pi r^2 h

Substituting the values:

Vcylinder=227×(0.5)2×16V_{\text{cylinder}} = \frac{22}{7} \times (0.5)^2 \times 16=227×0.25×16= \frac{22}{7} \times 0.25 \times 16=227×4= \frac{22}{7} \times 4=887= \frac{88}{7}≈12.57 cm3\approx 12.57 \text{ cm}^3

Step 2: Volume of the conical tips

The volume of a cone is given by:

V=13πr2hV = \frac{1}{3} \pi r^2 h

Substituting the values for one cone:

Vcone=13×227×(0.5)2×1.2V_{\text{cone}} = \frac{1}{3} \times \frac{22}{7} \times (0.5)^2 \times 1.2=13×227×0.25×1.2= \frac{1}{3} \times \frac{22}{7} \times 0.25 \times 1.2=13×227×0.3= \frac{1}{3} \times \frac{22}{7} \times 0.3=6.67= \frac{6.6}{7}≈0.94 cm3\approx 0.94 \text{ cm}^3

Since there are two cones, the total volume of the conical parts is:

Vcones=2×0.94=1.88 cm3V_{\text{cones}} = 2 \times 0.94 = 1.88 \text{ cm}^3

Step 3: Total Volume of the Pencil

Vtotal=Vcylinder+VconesV_{\text{total}} = V_{\text{cylinder}} + V_{\text{cones}}=12.57+1.88= 12.57 + 1.88=14.45 cm3= 14.45 \text{ cm}^3

Thus, the approximate volume of the pencil is 14.45 cm³.


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