Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 4 - Section 4.7 - Logarithmic Scales - 4.7 Exercises - Page 386: 21

Answer

(a) see prove below. (b) 106 dB

Work Step by Step

(a) Given $I=\frac{k}{d^2}$ and $B=10log\frac{I}{I_0}$, we have $B_1=10log\frac{I_1}{I_0}=10log\frac{k/d_1^2}{I_0}=10log\frac{k}{I_0d_1^2} =10(logk-logI_0-logd_1^2)=10(logk-logI_0)-20logd_1$ which in turn gives $10(logk-logI_0)=B_1+20logd_1$ Similarly, $B_2=10(logk-logI_0-logd_2^2)=10(logk-logI_0)-20logd_2 =B_1+20logd_1-20logd_2$ Thus, $B_2=B_1+20log\frac{d_1}{d_2}$ which proves the relationship. (b) Given $B_1=120,d_1=2,d_2=10$, use the above relationship, we have $B_2=120+20log\frac{2}{10}=106$ dB
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