Answer
See answers.
Work Step by Step
a. Equation 21-19 gives the resonant frequency.
$$f_o=\frac{1}{2 \pi}\sqrt{\frac{1}{LC}}$$
Take ratios.
$$\frac{f_{580}}{ f_{1600}}=\frac{580kHz}{1600kHz}=\frac{\frac{1}{2 \pi}\sqrt{\frac{1}{LC_{580}}}}{\frac{1}{2 \pi}\sqrt{\frac{1}{LC_{1600}}}}$$
$$C_{1600}=\frac{580^2}{1600^2}C_{580}=\frac{580^2}{1600^2}2800pF\approx 370pF$$
b. Calculate the inductance using the resonant frequency and the capacitance.
$$L=\frac{1}{4\pi^2 f^2 C}$$
$$L=\frac{1}{4\pi^2 (5.8\times10^5 Hz)^2 (2800\times10^{-12}F)}= 2.7\times10^{-5}H $$