To find the value of [tex]\( x \)[/tex], the time the train travels at 70 mph, we'll follow these steps:
1. Convert Mixed Number to Decimal:
First, we convert [tex]\(3 \frac{3}{4}\)[/tex] hours to a decimal.
[tex]\[
3 \frac{3}{4} = 3 + \frac{3}{4} = 3 + 0.75 = 3.75 \text{ hours}
\][/tex]
2. Calculate Distance at 80 mph:
The distance traveled at 80 mph for 3.75 hours can be calculated using the formula:
[tex]\[
\text{Distance} = \text{Speed} \times \text{Time}
\][/tex]
Here, the speed is 80 mph and the time is 3.75 hours.
[tex]\[
\text{Distance}_2 = 80 \text{ mph} \times 3.75 \text{ hours} = 300 \text{ miles}
\][/tex]
3. Determine Remaining Distance:
The total distance covered by the train is 405 miles. Thus, the remaining distance to be covered at 70 mph is:
[tex]\[
\text{Distance}_1 = 405 \text{ miles} - 300 \text{ miles} = 105 \text{ miles}
\][/tex]
4. Calculate Time at 70 mph:
Now, we need to find the time [tex]\( x \)[/tex] that the train travels at 70 mph to cover the remaining 105 miles. Using the formula:
[tex]\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\][/tex]
For the remaining distance:
[tex]\[
x = \frac{105 \text{ miles}}{70 \text{ mph}} = 1.5 \text{ hours}
\][/tex]
5. Conclusion:
The train travels at 70 mph for [tex]\( x = 1.5 \)[/tex] hours.
Thus, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = 1.5 \text{ hours}
\][/tex]
