To determine how long it will take for a signal to travel from Earth to Mars at a given distance and signal speed, we can use the formula:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \][/tex]
Given:
- The minimum distance between Earth and Mars is [tex]\(5.46 \times 10^7 \)[/tex] kilometers.
- The speed of the signal is [tex]\(4 \times 10^5 \)[/tex] kilometers per second.
We can plug these values into the formula:
[tex]\[ \text{Time} = \frac{5.46 \times 10^7 \text{ km}}{4 \times 10^5 \text{ km/s}} \][/tex]
Carrying out the division:
[tex]\[ \text{Time} = \frac{5.46 \times 10^7}{4 \times 10^5} \][/tex]
To simplify the division, we can divide the coefficients (5.46 by 4) and the powers of 10:
[tex]\[ \frac{5.46}{4} = 1.365 \][/tex]
For the powers of 10:
[tex]\[ \frac{10^7}{10^5} = 10^{7-5} = 10^2 = 100 \][/tex]
Now multiply the simplified result:
[tex]\[ 1.365 \times 100 = 136.5 \][/tex]
Thus, it will take the signal 136.5 seconds to travel from Earth to Mars when they are at the minimum distance.
