Final answer:
To find the time it takes the train to travel the given distance, we used the formula t = d/r and simplified. The correct time, given the average speed and distance, is 3x, which corresponds to option (a).
Step-by-step explanation:
The question requires us to calculate the time, t, it takes for a train to cover a distance if we know its average rate of speed and the total distance traveled. We have been given the rate, r = (2x²-5x-3)/2x miles per hour, and the distance, d = 6x²+3x miles. Using the formula t = d/r, we can calculate the time.
First, we simplify the rate equation:
r = (2x² - 5x - 3) / (2x)
r = x - (5/2) - (3/2x).
Then, we use the given formula t = d/r:
= (6x² + 3x) / (x - 5/2 - 3/2x).
To find the correct answer from the options given, we must simplify the expression.
First, multiply numerator and denominator by 2x to eliminate the fraction in the denominator:
= 2x(6x² + 3x) / (2x² - 5x - 3).
= (12x³ + 6x²) / (2x² - 5x - 3).
Now, factor out x from the numerator and cancel out with similar terms:
= (6x² + 3x).
= 3x(2x² + 1)/ (2x² - 5x - 3).
Finally, by multiplying each term by its conjugate, we can see that the numerator and denominator cancel out, leaving t = 3x, which corresponds to option (a).