Write 868 million in the form $\alpha \times 10^n$, where $n \in \mathbb{Z}$ and $1 \leq \alpha < 10$, $\alpha \in \mathbb{R}$ - Junior Cycle Mathematics - Question 3 - 2017

Given that it takes between 1 to 3 seconds for the radio signal to travel 380,000 km:

Average speed can be calculated using the formula:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

For 1 second:

$\text{Speed} = \frac{380,000 \text{ km}}{1 \text{ second}} = 380,000 \text{ km/s}$

For 3 seconds:

$\text{Speed} = \frac{380,000 \text{ km}}{3 \text{ seconds}} \approx 126,667 \text{ km/s}$

To convert seconds to minutes, we multiply by 60:

• Speed for 1 second: $\text{Speed} = 380,000 \text{ km/s} \times 60 \text{ s/min} = 22,800,000 \text{ km/min} = 2.28 \times 10^7 \text{ km/min}$

• Speed for 3 seconds: $\text{Speed} \approx 126,667 \text{ km/s} \times 60 \text{ s/min} \approx 7,600,000 \text{ km/min} = 7.60 \times 10^6 \text{ km/min}$

To find the time taken for a radio signal to travel 868 million km at the speed computed in part (b):

Using the speed obtained for 1 second (average speed):

$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$

Distance = 868 million km

Using the speed from the first part:

$\text{Time} = \frac{868,000,000 \text{ km}}{22,800,000 \text{ km/min}} \approx 38.1 \text{ minutes}$

Using the speed from the second part:

$\text{Time} = \frac{868,000,000 \text{ km}}{7,600,000 \text{ km/min}} \approx 114.3 \text{ minutes}$

Concluding that it takes approximately 49.6 minutes using 1.75 \times 10^7 as a reasonable speed from the average range.