Jake and Sara each drive 270 miles to attend a conference. Jake drives at an average
speed that is 15 mi/h slower than Sara’s average speed. It takes Jake 1.5 hours longer
than Sara to drive the 270 miles. How long does it Jake to make the trip?
since time = distance/speed, If Jake's speed is x,
270/x = 270/(x+15) + 1.5
x = 45
So, it takes Jake 270/45 = 6 hours
To solve this problem, we can set up a system of equations.
Let's say Sara's average speed is x mi/h. According to the problem, Jake's average speed is 15 mi/h slower than Sara's, so Jake's average speed is (x - 15) mi/h.
We can use the formula speed = distance/time to express the time it takes for each person to drive the 270 miles.
For Sara: time = distance / speed
For Jake: time = distance / speed
Given that it takes Jake 1.5 hours longer than Sara to drive the 270 miles, we can set up the following equation:
(time taken by Jake) - (time taken by Sara) = 1.5 hours
Substituting the equations for time, we get:
(distance / Jake's speed) - (distance / Sara's speed) = 1.5
Plugging in the values for distance and speed, we get:
(270 / (x - 15)) - (270 / x) = 1.5
Now, we can solve this equation to find the value of x.
To do that, we can cross multiply:
(270 * x) - (270 * (x - 15)) = 1.5 * x * (x - 15)
Simplifying the equation gives us:
270x - 270(x - 15) = 1.5x(x - 15)
270x - 270x + 4050 = 1.5x^2 - 22.5x
Rearranging and simplifying further:
1.5x^2 - 22.5x - 4050 = 0
Now, we have a quadratic equation and we can solve it using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In our equation, a = 1.5, b = -22.5, and c = -4050.
Substituting these values into the quadratic formula:
x = (-(-22.5) ± sqrt((-22.5)^2 - 4(1.5)(-4050))) / (2 * 1.5)
Simplifying further:
x = (22.5 ± sqrt(506.25 + 24300)) / 3
x = (22.5 ± sqrt(24806.25)) / 3
Now, we can calculate the two possible values for x using a calculator:
x ≈ (22.5 + sqrt(24806.25)) / 3 ≈ 53.20 mi/h
or
x ≈ (22.5 - sqrt(24806.25)) / 3 ≈ -44.87 mi/h (disregarding this negative value as it is not possible for speed)
Therefore, Sara's average speed is approximately 53.20 mi/h, and we can now calculate the time it takes for Jake to make the trip using his average speed:
Jake's speed = Sara's speed - 15 mi/h ≈ 53.20 mi/h - 15 mi/h ≈ 38.20 mi/h
Now, we can use the formula time = distance / speed:
Jake's time = 270 miles / 38.20 mi/h ≈ 7.06 hours
So, it takes Jake approximately 7.06 hours (or around 7 hours and 4 minutes) to make the trip.