Chapter 11, Problem 51RE

a.

To determine

To find: The distance between sun and earth when the latter is closest to the sun in miles.

Answer to Problem 51RE

The distance between sun and earth when the latter is closest to sun is $91\cdot 42\times {10}^6\text{ miles}$ .

Explanation of Solution

Given information:

Length of major axis is $186\times {10}^6$ and

  $\begin{array}{l}2a=186\times {10}^6\text{ mi}\\ \text{a = 93}\times {\text{10}}^6\text{ mi}\end{array}$

Concept Used:

The distance between a focus and its nearest vertex when eccentricity is given is $a\left(1-e\right)$

Calculation:

Distance between earth and sun when earth is closes to the sun is $a\left(1-e\right)$

  $\begin{array}{l}=93\times {10}^6\left(1-0\cdot 017\right)\\ =93\times {10}^6\left(0\cdot 983\right)\\ =91\cdot 42\times {10}^6\text{ mi}\end{array}$

Conclusion:

Hence,

Distance between earth and sun when earth is

Closes to sun $=91\cdot 42\times {10}^6\text{ miles}$

b.

To determine

To find: The distance between sun and earth when the latter is farthest to the sun in miles.

Answer to Problem 51RE

The distance between sun and earth when the latter is farthest to the sun is $94\cdot 56\times {10}^6\text{ miles}$

Explanation of Solution

Given information:

Length of major axis is $186\times {10}^6$ and

  $\begin{array}{l}2a=186\times {10}^6\text{ mi}\\ \text{a = 93}\times {\text{10}}^6\text{ mi}\end{array}$

Concept Used:

The distance between a focus and its farthest vertex is given is $a\left(1+e\right)$

Calculation:

Distance between earth and sun is farthest to sun is $a+ae=a\left(1+e\right)$

  $\begin{array}{l}=93\times {10}^6\left(1+0\cdot 017\right)\\ =93\times {10}^6\left(1\cdot 017\right)\\ =94\cdot 581\times {10}^6\text{ miles}\end{array}$

Conclusion:

Hence,

Distance between earth and sun when earth is

Farthest to sun $=94\cdot 581\times {10}^6\text{ miles}$