Chapter 7.1, Problem 66PS

Solution

i.

To state: The approximate the distance of Mars from the sun using the Titius-Bode rule if the value of n is 4 for Mars. The Titius-Bode rule $d_n=0.3{\left(2\right)}^{n-2}+0.4$ , where n is a positive integer representing the position of the planet from the sun.

The resultant distance is 1.6.

Given information:

The given rule is $d_n=0.3{\left(2\right)}^{n-2}+0.4$ .

Explanation:

Consider the rule $d_n=0.3{\left(2\right)}^{n-2}+0.4$:

Substitute the value of n as 4 in the given rule,

  $\begin{array}{c}{d}_n=0.3{\left(2\right)}^{4-2}+0.4\\ =0.3{\left(2\right)}^2+0.4\\ =1.2+0.4\\ =1.6\end{array}$

Therefore, the distance of Mars from the sun is 1.6.

ii.

To state: The distance of Mar from the sun in kilometers if one astronomical unit is equal to about $149,600,000$ kilometers.

The resultant distance is $239,360,000$ .

Given information:

One astronomical unit is equal to about $149,600,000$ kilometers.

Explanation:

From part (a) the distance of Mars from Sun is 1.6 astronomical units,

The distance of Mars from Sun in kilometers is:

  $\left(1.6\right)149,600,000=239,360,000$

Therefore, the distance of Mars from Sun is $239,360,000$ kilometers.

iii.

To graph: The sequence $d_n=0.3{\left(2\right)}^{n-2}+0.4$ .

Given information:

The given sequence is $d_n=0.3{\left(2\right)}^{n-2}+0.4$ .

Explanation:

Consider the given sequence: $d_n=0.3{\left(2\right)}^{n-2}+0.4$

Find the first few terms of the sequence by substituting the values of n as:

  $1,2,3,4,5,6,7$

  $\begin{array}{c}{d}_1=0.3{\left(2\right)}^{1-2}+0.4,d_1=0.55\\ d_2=0.3{\left(2\right)}^{2-2}+0.4,d_2=0.7\\ d_3=0.3{\left(2\right)}^{3-2}+0.4,d_3=1\\ d_4=0.3{\left(2\right)}^{4-2}+0.4,d_4=1.6\\ d_5=0.3{\left(2\right)}^{5-2}+0.4,d_5=2.8\\ d_6=0.3{\left(2\right)}^{6-2}+0.4,d_6=5.2\\ d_7=0.3{\left(2\right)}^{7-2}+0.4,d_7=10\end{array}$

Now plot the points on the graph:

  $\left(1,0.55\right),\left(2,0.7\right),\left(3,1\right),\left(4,1.6\right),\left(5,2.8\right),\left(6,5.2\right),\left(7,10\right)$

  

Interpretation: The graph of $d_n=0.3{\left(2\right)}^{n-2}+0.4$ is increasing.