Radio Dial Derivative Problem: Figure 6-2m shows an old AM radio dial. As you can see

Chapter 6, Problem 61

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Radio Dial Derivative Problem: Figure 6-2m shows an old AM radio dial. As you can see, the distances between numbers decrease as the frequency increases. If you study the theory behind the tuning of radios, you will learn that the distance from the left end of the dial to a particular frequency varies logarithmically with the frequency. That is, d( f )= a + b ln f where d( f ) is the number of centimeters from the number 53 to the frequency number f on the dial, and where a and b stand for constants. a. Solve for the constants a and b to find the particular equation for this logarithmic function. b. Use the equation you found in part a to make a table of values of d( f ) for each frequency shown in Figure 6-2m. Then measure their distances with a ruler to the nearest 0.1 cm.If your calculated and measured answers donot agree, go back and fix your errors. c. Write an equation for ( f ). In the table frompart b, put a new column that shows theinstantaneous rates of change of distancewith respect to frequency.d. The numbers on the dial in Figure 6-2m are given in tens of kilohertz. (One hertz equalsone cycle per second.) What are the units of ( f )? e. Do the values of ( f ) increase or decrease as f increases? Explain how this fact isconsistent with the way the numbers arespaced on the dial.