Chapter 7.3, Problem 86PPS

a.

To determine

To calculate the number of steps that has been taken in 2-direction walk if there are 4096 possible walks.

Answer to Problem 86PPS

12

Explanation of Solution

Given:

The number of possible random walks w of n steps where you choose one of d directions at each step is $w=d^n$

  $d=2$ , $w=4096$

Formula used:

  $x^{\frac{a}{b}}=\sqrt[b]{x^a}$

Calculation:

The number of possible walks w of n steps with d directions is $w=d^n$ .

Given that $d=2$ , $w=4096$

Substituting in the given expression,

  $4096=2^n$

Here, 4096 can be written as $2^{12}$

  $\begin{array}{l}\Rightarrow 2^{12}=2^n\\ \Rightarrow n=12[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Conclusion:

Therefore, the number of steps in 2-direction walk is 12.

b.

To determine

To calculate the number of steps that has been taken in 4-direction walk if there are 65536 possible walks.

Answer to Problem 86PPS

8

Explanation of Solution

Given:

The number of possible random walks w of n steps where you choose one of d directions at each step is $w=d^n$

  $d=4$ , $w=65536$

Formula used:

  $x^{\frac{a}{b}}=\sqrt[b]{x^a}$

Calculation:

The number of possible walks w of n steps with d directions is $w=d^n$ .

Given that $d=4$ , $w=65536$

Substituting in the given expression,

  $65536=4^n$

Here, 65536 can be written as $4^8$

  $\begin{array}{l}\Rightarrow 4^8=4^n\\ \Rightarrow n=8[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Conclusion:

Therefore, the number of steps in 4-direction walk is 8.

c.

To determine

To calculate the number of direction in each step.

Answer to Problem 86PPS

7

Explanation of Solution

Given:

The number of possible random walks w of n steps where you choose one of d directions at each step is $w=d^n$

  $n=7$ , $w=2187$

Formula used:

  $x^{\frac{a}{b}}=\sqrt[b]{x^a}$

Calculation:

The number of possible walks w of n steps with d directions is $w=d^n$ .

Given that $n=7$ , $w=2187$

Substituting in the given expression,

  $2187=d^7$

  $\begin{array}{l}\Rightarrow d=\sqrt[7]{2187}\\ \Rightarrow d=3\end{array}$

Conclusion:

Therefore, there can be 3 directions in each step.