Answer :
Let's solve the problem step-by-step:
### Given:
- Speed of light: 186,000 miles per second
- Conversion factors:
- 1 mile = 5280 feet
- 1 foot = 12 inches
- 1 inch = 2.54 centimeters
### Steps to Convert the Speed of Light to Centimeters per Second:
1. Convert miles per second to feet per second:
[tex]\[
\text{Speed in feet per second} = \text{Speed in miles per second} \times \text{feet per mile}
\][/tex]
[tex]\[
\text{Speed in feet per second} = 186,000 \, \text{miles/second} \times 5280 \, \text{feet/mile}
\][/tex]
[tex]\[
\text{Speed in feet per second} = 982,080,000 \, \text{feet/second}
\][/tex]
2. Convert feet per second to inches per second:
[tex]\[
\text{Speed in inches per second} = \text{Speed in feet per second} \times \text{inches per foot}
\][/tex]
[tex]\[
\text{Speed in inches per second} = 982,080,000 \, \text{feet/second} \times 12 \, \text{inches/foot}
\][/tex]
[tex]\[
\text{Speed in inches per second} = 11,784,960,000 \, \text{inches/second}
\][/tex]
3. Convert inches per second to centimeters per second:
[tex]\[
\text{Speed in centimeters per second} = \text{Speed in inches per second} \times \text{centimeters per inch}
\][/tex]
[tex]\[
\text{Speed in centimeters per second} = 11,784,960,000 \, \text{inches/second} \times 2.54 \, \text{cm/inch}
\][/tex]
[tex]\[
\text{Speed in centimeters per second} = 29,933,798,400 \, \text{cm/second}
\][/tex]
4. Express the final answer in scientific notation:
[tex]\[
29,933,798,400 \, \text{cm/second} = 2.99337984 \times 10^{10} \, \text{cm/second} \approx 3.00 \times 10^{10} \, \text{cm/second}
\][/tex]
Thus, the speed of light in centimeters per second is approximately [tex]\(3.00 \times 10^{10} \, \text{cm/second}\)[/tex].
Among the given options, the closest standard scientific notation is [tex]\( \boxed{3.15 \times 10^{10} \text{cm/s}} \)[/tex]. However, if we strictly follow our precise calculation and acceptable rounding, the scientific notation should look more like [tex]\( \boxed{3.00 \times 10^{10} \text{cm/s}} \)[/tex].
### Given:
- Speed of light: 186,000 miles per second
- Conversion factors:
- 1 mile = 5280 feet
- 1 foot = 12 inches
- 1 inch = 2.54 centimeters
### Steps to Convert the Speed of Light to Centimeters per Second:
1. Convert miles per second to feet per second:
[tex]\[
\text{Speed in feet per second} = \text{Speed in miles per second} \times \text{feet per mile}
\][/tex]
[tex]\[
\text{Speed in feet per second} = 186,000 \, \text{miles/second} \times 5280 \, \text{feet/mile}
\][/tex]
[tex]\[
\text{Speed in feet per second} = 982,080,000 \, \text{feet/second}
\][/tex]
2. Convert feet per second to inches per second:
[tex]\[
\text{Speed in inches per second} = \text{Speed in feet per second} \times \text{inches per foot}
\][/tex]
[tex]\[
\text{Speed in inches per second} = 982,080,000 \, \text{feet/second} \times 12 \, \text{inches/foot}
\][/tex]
[tex]\[
\text{Speed in inches per second} = 11,784,960,000 \, \text{inches/second}
\][/tex]
3. Convert inches per second to centimeters per second:
[tex]\[
\text{Speed in centimeters per second} = \text{Speed in inches per second} \times \text{centimeters per inch}
\][/tex]
[tex]\[
\text{Speed in centimeters per second} = 11,784,960,000 \, \text{inches/second} \times 2.54 \, \text{cm/inch}
\][/tex]
[tex]\[
\text{Speed in centimeters per second} = 29,933,798,400 \, \text{cm/second}
\][/tex]
4. Express the final answer in scientific notation:
[tex]\[
29,933,798,400 \, \text{cm/second} = 2.99337984 \times 10^{10} \, \text{cm/second} \approx 3.00 \times 10^{10} \, \text{cm/second}
\][/tex]
Thus, the speed of light in centimeters per second is approximately [tex]\(3.00 \times 10^{10} \, \text{cm/second}\)[/tex].
Among the given options, the closest standard scientific notation is [tex]\( \boxed{3.15 \times 10^{10} \text{cm/s}} \)[/tex]. However, if we strictly follow our precise calculation and acceptable rounding, the scientific notation should look more like [tex]\( \boxed{3.00 \times 10^{10} \text{cm/s}} \)[/tex].
