Question

The speed of light is about \[2\times {{10}^{5}}\] miles per second. On an average, it takes light about 500 seconds to travel from the sun to earth. What is the average distance from Earth to the sun? Write your answer in scientific notation.

Answer 1

Hint: In this type of question we have to consider the concept of representing a number in scientific notation. We know that any number can be converted into the form of scientific notation i.e. in the form of \[N\times {{10}^{a}}\], where \[1\le \left| N \right|<10\] and ‘\[a\]’ is an integer. To make any number in scientific notation first move the decimal to right or left to create a number \[N\], \[1\le \left| N \right|<10\]. Then determine the value of \[a\] which is equal to the number of times we moved the decimal, if left then \[a\] is positive and if right then \[a\] is negative.

Complete step by step solution:
Now, we have to find the average distance from Earth to the sun and then we have to express the answer in scientific notation.
We have given that,
Speed of light = \[2\times {{10}^{5}}\] miles per second
Time taken to travel from sun to Earth = 500 seconds
\[\begin{align}
  & \Rightarrow \text{Average Distance = Speed }\times \text{ Time taken} \\
 & \Rightarrow \text{Average Distance = 2}\times \text{1}{{\text{0}}^{5}}\times 500\text{ miles} \\
 & \Rightarrow \text{Average Distance = 1000}\times \text{1}{{\text{0}}^{5}}\text{ miles} \\
\end{align}\]
Hence, the average distance from Earth to the sun is \[\text{1000}\times \text{1}{{\text{0}}^{5}}\text{ miles}\]
Now we have to express it in scientific notation.
 To make any number in scientific notation first move the decimal to right or left to create a number \[N\], \[1\le \left| N \right|<10\]. Then determine the value of \[a\] which is equal to the number of times we moved the decimal, if left then \[a\] is positive and if right then \[a\] is negative.
Hence, we can express \[\text{1000}\] as \[1\times {{10}^{3}}\]
\[\begin{align}
  & \Rightarrow \text{Average Distance = 1}\times \text{1}{{\text{0}}^{3}}\times \text{1}{{\text{0}}^{5}}\text{ miles} \\
 & \Rightarrow \text{Average Distance = 1}\times \text{1}{{\text{0}}^{8}}\text{ miles} \\
\end{align}\]
Hence in scientific notation, average distance from Earth to the sun is \[\text{1}{{\text{0}}^{8}}\text{ miles}\]

Hence we can write, the average distance from Earth to the sun is \[\text{1000}\times \text{1}{{\text{0}}^{5}}\text{ miles}\] which in scientific notation can be expressed as \[\text{1}{{\text{0}}^{8}}\text{ miles}\]

Note:
In this type of question students may make mistakes if there are too many zeros in an expression students tend to miss some of them, getting incorrect answers. Students have to take care if there are too many zeros then we can write the number in scientific notation to decrease the error rate.