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Chapter 4: Problem 74

A tapered shaft has a diameter of 5 centimeters at the small end and is 15 centimeters long (see figure). The taper is \(3^{\circ} .\) Find the diameter \(d\) of the large end of the shaft.

Short Answer

Expert verified
The diameter of the large end of the shaft is approximately 6.28 centimetres.

Step by step solution

01

Convert angle to radians

The angle of taper is given in degrees. While calculating trigonometric functions, radians are often used instead of degrees. Convert the angle of taper(3 degrees) to radian. Multiply the degree by \(\pi /180\) to convert it into radian. Therefore, \(3^{\circ} *(\pi /180) = 0.0523599 \) radians.
02

Set up the trigonometric relationship

As the shadow taper creates a right triangle, consider taper angle \( \theta \) on the right triangle. Then, opposite side to the taper angle is \(\frac{d-d_1}{2}\) (half of the difference of the diameters), and adjacent side is the length \( L \) of the shaft. Here, \(d_1\) (diameter of the small end) is 5 cm and \( L \) is 15 cm. The tangent of the angle is the ratio of the opposite strength to the adjacent strength, that is \( tan(\theta) = \frac{d-d_1}{2L} \) . Substituting the values of \( \theta, d_1 \) and \( L \), we get \( tan(0.0523599) = \frac{d-5}{30} \) .
03

Solve for the unknown diameter

Now solve the equation from step 2 for \( d \) . By rearranging the equation we get \( d = 2*L*tan(\theta) + d_1 \) . Now, plug the values into the equation and get the value of diameter \( d \) , yields \( d = 2*15*tan(0.0523599) + 5 = 6.28317 \, cm \) .

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