A family drove 1080 miles to their vacation lodge. Because of increased traffic density, their average speed on the return trip was decreased by 6 miles per hour and the trip took 2.5 hours longer. Determine their average speed on the way to the lodge.
since time = distance/speed,
1080/s + 5/2 = 1080/(s-6)
s = 54
So, speed out = 54, sped back = 48
2160/(1080/54 + 1080/48) = 50.8 mph avg speed
Let's assume the average speed of the family on their way to the lodge is "x" miles per hour.
We know that the distance to the lodge is 1080 miles.
On the return trip, their average speed decreased by 6 miles per hour, which means their speed became (x - 6) miles per hour.
We also know that the return trip took 2.5 hours longer than the trip to the lodge. Let's call the time taken on the way to the lodge as "t" hours.
On the way to the lodge:
Distance = Speed * Time
1080 miles = x miles per hour * t hours
t = 1080 / x
On the return trip:
Distance = Speed * Time
1080 miles = (x - 6) miles per hour * (t + 2.5) hours
1080 = (x - 6) * (t + 2.5)
1080 = (x - 6) * (1080 / x + 2.5)
1080x = (x - 6) * (1080 + 2.5x)
1080x = 1080x + 2.5x^2 - 6480 - 15x
0 = 2.5x^2 - 15x - 6480
To solve this quadratic equation, we can either factorize it or use the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the values from the equation 2.5x^2 - 15x - 6480 = 0:
a = 2.5
b = -15
c = -6480
Plugging these values into the quadratic formula:
x = (-(-15) ± √((-15)^2 - 4 * 2.5 * -6480)) / (2 * 2.5)
x = (15 ± √(225 + 64800)) / 5
x = (15 ± √65025) / 5
x = (15 ± 255) / 5
Since the average speed cannot be negative, we ignore the negative value:
x = (15 + 255) / 5
x = 270 / 5
x = 54
Therefore, their average speed on the way to the lodge was 54 miles per hour.
To determine the average speed on the way to the lodge, we need to consider the formula:
Average Speed = Total Distance / Total Time
Let's break down the problem to find the missing information step by step:
1. Calculate the total distance of the round trip:
Since the family drove 1080 miles to the lodge, the total distance of the round trip is 2 * 1080 = 2160 miles.
2. Let's assume the average speed on the way to the lodge as 'x' miles per hour.
3. Calculate the total time for the return trip:
On the return trip, the average speed decreased by 6 miles per hour. So, the average speed for the return trip will be (x - 6) miles per hour.
The time taken for the return trip can be calculated using the formula: Time = Distance / Speed.
So, the time taken for the return trip is 1080 miles / (x - 6) miles per hour.
4. Calculate the total time for the trip:
The time taken for the trip to the lodge is 1080 miles / x miles per hour.
5. Given that the return trip took 2.5 hours longer, we can set up the following equation:
1080 miles / x miles per hour + 2.5 hours = 1080 miles / (x - 6) miles per hour
Now, we can solve the equation to find the average speed on the way to the lodge:
1080 / x + 2.5 = 1080 / (x - 6)
First, we can simplify the equation by clearing the fractions:
(x - 6) * (1080 / x + 2.5) = 1080
(1080 * (x - 6)) / x + 2.5 * (x - 6) = 1080
(1080 * (x - 6)) / x + 2.5x - 15 = 1080
Next, we can cross-multiply to eliminate the fractions:
1080 * (x - 6) + 2.5x * x - 6 * 2.5x - 6 * 15 = 1080 * x
1080x - 6480 + 2.5x^2 - 15x - 45 = 1080x
Rearranging the equation:
2.5x^2 - 15x - 6930 = 0
Now, we can solve this quadratic equation for the value of 'x' using factoring, completing the square, or the quadratic formula.
Using the quadratic formula:
x = (-(-15) ± sqrt((-15)^2 - 4(2.5)(-6930))) / (2 * 2.5)
Simplifying:
x = (15 ± sqrt(225 + 4(2.5)(6930))) / 5
x = (15 ± sqrt(225 + 69300)) / 5
x = (15 ± sqrt(69525)) / 5
x ≈ (15 ± 263.87) / 5
x can be approximately equal to:
x ≈ (15 + 263.87) / 5 ≈ 55.77 miles per hour (Discarding the negative value)
Therefore, the average speed on the way to the lodge is approximately 55.77 miles per hour.