The distance of the Moon from Earth is approximately 384400 km . If a spacecraft travels at an average speed of $$ 58536 \mathrm{~km} / \mathrm{h} $$ to the Moon and then at an average speed of $$ 52000 \mathrm{~km} / \mathrm{h} $$ back to Earth, calculate the total time taken for the trip to the Moon and back home.

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To calculate the total time taken for the trip to the Moon and back home, we need to find the time taken for each leg of the journey and then add them together.

Given:
- Distance to the Moon = 384400 km
- Average speed to the Moon = 58536 km/h
- Average speed back to Earth = 52000 km/h

1. Calculate the time taken to travel to the Moon:
$$ \text{Time to the Moon} = \frac{\text{Distance to the Moon}}{\text{Average speed to the Moon}} $$

2. Calculate the time taken to travel back to Earth:
$$ \text{Time back to Earth} = \frac{\text{Distance to the Moon}}{\text{Average speed back to Earth}} $$

3. Calculate the total time taken for the trip:
$$ \text{Total time} = \text{Time to the Moon} + \text{Time back to Earth} $$

Let's calculate the time taken for each leg of the journey and then find the total time.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{384400}{58536}$$
- step1: Reduce the fraction:
  $$\frac{48050}{7317}$$
Calculate or simplify the expression $$ 384400 / 52000 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{384400}{52000}$$
- step1: Reduce the fraction:
  $$\frac{961}{130}$$
Calculate or simplify the expression $$ 6.566899 + 7.3\dot{9}2307\dot{6} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$6.566899+7.3\overline{923076}$$
- step1: Convert the decimal into a fraction:
  $$6.566899+\frac{961}{130}$$
- step2: Convert the expressions:
  $$\frac{6566899}{1000000}+\frac{961}{130}$$
- step3: Reduce fractions to a common denominator:
  $$\frac{6566899\times 13}{1000000\times 13}+\frac{961\times 100000}{130\times 100000}$$
- step4: Multiply the numbers:
  $$\frac{6566899\times 13}{13000000}+\frac{961\times 100000}{130\times 100000}$$
- step5: Multiply the numbers:
  $$\frac{6566899\times 13}{13000000}+\frac{961\times 100000}{13000000}$$
- step6: Transform the expression:
  $$\frac{6566899\times 13+961\times 100000}{13000000}$$
- step7: Multiply the numbers:
  $$\frac{85369687+961\times 100000}{13000000}$$
- step8: Multiply the numbers:
  $$\frac{85369687+96100000}{13000000}$$
- step9: Add the numbers:
  $$\frac{181469687}{13000000}$$
The time taken to travel to the Moon is approximately 6.5669 hours.
The time taken to travel back to Earth is approximately 7.3 hours.
The total time taken for the trip to the Moon and back home is approximately 13.9592 hours.
The total time taken for the trip to the Moon and back home is approximately 13.96 hours.

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