Related questions with answers

Different numbers x, y and z are the first three terms of a geometric progression with common ratio r and also the first, second and fifth terms of an arithmetic progression. (a) Find the value of r. (b) Find which term of the arithmetic progression will next be equal to a term of the geometric progression.

In a box, Lynn has b batteries of which d are dead. She tests them randomly and one by one. Every time that a good battery is drawn, she will retum it to the box; every time that a dead battery is drawn, she will replace it by a good one.

(a) Determine the expected value of the number of good batteries in the box after n of them are checked.

(b) Determine the probability that on the nth draw Lynn draws a good battery. Hint: Let $X_n$ be the number of good batteries in the box after n of them are checked. Show that

E\left(X_n \mid X_{n-1}\right)=1+\left(1-\frac{1}{b}\right) X_{n-1} \text {. }

Then, by computing the expected value of this random variable, find a recursive relation between $E\left(X_n\right)$ and $E\left(X_{n-1}\right)$ . Use this relation and induction to prove that

E\left(X_n\right)=b-d\left(1-\frac{1}{b}\right)^n .

Note that $n$ should approach $\infty$ to get $E\left(X_n\right)=b$ . For part (b), let $E_n$ be the event that on the nth draw she gets a good battery. By conditioning on $X_{n-1}$ prove that $P\left(E_n\right)=E\left(X_{n-1}\right) / b$ .

What is the name for the ejected lava fragments that include pea- to walnut-sized pieces?

Question

At f = 100 MHz, show that silver $\left(\sigma=6.1 \times 10^{7} \mathrm{S} / \mathrm{m}, \mu_{r}=1, \varepsilon_{r}=1\right)$ is a good conductor, while rubber $\left(\sigma=10^{-15} \mathrm{S} / \mathrm{m}, \mu_{r}=1, \varepsilon_{r}=3.1\right)$ is a good insulator.

Solution

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We are given a frequency of $f=100 ~\mathrm{MHz}$ , and the parameters of two materials:

Silver $(\sigma = 6.1 \cdot 10^7 \mathrm{~\frac{S}{m}}, \epsilon_r = 1, \mu_r = 1)$
Rubber $(\sigma = \cdot 10^{-15} \mathrm{~\frac{S}{m}}, \epsilon_r = 1, \mu_r = 3.1)$