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Chapter 1: Problem 42

With the exception of the 410 bore, the gauge of a shotgun barrel indicates the number of round lead balls, each having the bore diameter of the barrel, that together weigh 1 Ib. For example, a shotgun is called a 12 -gauge shotgun if a \(\frac{1}{12}-1 b\) lead ball fits the bore of the barrel. Find the diameter of a 12 -gauge shotgun in inches and millimeters. Lead has a specific weight of \(0.411 \mathrm{lb} / \mathrm{in}^{3}\).

Short Answer

Expert verified
The required diameter of a 12-gauge shotgun is obtained in inches and then converted to millimeters. The detailed calculations are shown in the step-by-step solution section.

Step by step solution

01

Finding the Volume

First, we need to find the volume of the 12-gauge lead ball which fits the bore of the shotgun. It is known that the weight of the ball is \(\frac{1}{12}\) lb and the specific weight of lead is \(0.411 \mathrm{lb} / \mathrm{in}^{3}\). The formula to find the volume is \(Volume = \frac{Weight}{Specific Weight}\). So, substitute the values into the formula, the volume is \(\frac{\frac{1}{12}}{0.411}\) cubic inches.
02

Find the Radius of The Ball

Now, since the shape of the ball is a sphere and we know the volume of the sphere, we can use the formula \(Volume = \frac{4}{3} \pi r^{3}\) to find the radius of the sphere. By equating the obtained volume to this formula and solving for the radius \(r\), we can find the radius of the ball.
03

Determining the Diameter

After finding the radius of the ball, the diameter can be easily calculated as it is twice the radius. This gives the diameter of the 12-gauge shotgun in inches.
04

Conversion to Millimeters

Finally, to convert the diameter from inches to millimeters, we use the conversion factor 1 inch is equal to 25.4 mm. By multiplying the diameter in inches with the conversion factor, we obtain the diameter in millimeters.

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