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.
helppp?!?
1080=vavg*time
1080=(vavg-6)(time+2.5)
from the first:
time=1080/vavg putting that into the second..
1080=(vavg-6)(1080/vavg +2.5)
1080=1080-6*1080/vavg + 2.5vavg-15
0=648+2.4 vavg^2 -15vavg
put it in standard form, use the quadratic formula. check my work.
=
To determine their average speed on the way to the lodge, we can use the formula:
Average Speed = Total Distance / Total Time.
Let's break down the information given:
1. The family drove 1080 miles to their vacation lodge.
2. On the return trip, their average speed was decreased by 6 miles per hour.
3. The return trip took 2.5 hours longer than the trip to the lodge.
Now, let's calculate the total time for each trip:
For the trip to the lodge:
Average Speed = Total Distance / Total Time
Let the average speed on the way to the lodge be represented by "x" miles per hour.
So, the time for the trip to the lodge would be:
Time (to the lodge) = Distance / Speed
t1 = 1080 / x
For the return trip:
The average speed on the return trip is decreased by 6 miles per hour, so it would be (x - 6) miles per hour.
The time for the return trip would be:
Time (return trip) = Distance / Speed
t2 = 1080 / (x - 6)
Based on the given information, we know that the return trip took 2.5 hours longer than the trip to the lodge. We can express this as an equation:
t2 = t1 + 2.5
Now, we can substitute the calculated times into the equation:
1080 / (x - 6) = 1080 / x + 2.5
To determine the value of x, let's solve this equation step by step:
1. Multiply both sides of the equation by x(x - 6) to eliminate the denominators:
1080x = 1080(x - 6) + 2.5x(x - 6)
2. Simplify and expand the equation:
1080x = 1080x - 6480 + 2.5x^2 - 15x
3. Rearrange the equation and combine the like terms:
2.5x^2 - 15x = -6480
4. Simplify the equation by dividing both sides by 2.5:
x^2 - 6x + 2592 = 0
5. Now, we can solve the quadratic equation either by factoring, completing the square, or using the quadratic formula. In this case, the quadratic equation factors as:
(x - 36)(x - 72) = 0
By setting each factor equal to zero, we get two possible values for x: x = 36 and x = 72.
However, we need to find the average speed on the way to the lodge, so we consider the value x = 72. This gives us an average speed of 72 miles per hour.
Therefore, the family's average speed on the way to the lodge was 72 miles per hour.