A compact car is driven 144mi. If it had traveled 4mph faster, it could have made the trip in 1/2 hour less time. What was the speed of the car?
Recall that rate is distance traveled over time, or
r = d/t
Solving for distance, we have
d = rt
Let r = speed of car
Let t = time it takes to travel 144 miles
Thus,
rt = 144 ; equation (1)
If it had traveled 4 mph faster, time it takes to travel the same distance would be less by 1/2 hr. In equation,
rt = (r + 4)(t - 1/2)
Solving,
rt = rt - 0.5r + 4t - 2
0 = 0 - 0.5r + 4t - 2
r = 8t - 4
Substituting this to equation (1),
rt = 144
(8t - 4)(t) = 144
8t^2 - 4t - 144 = 0
2t^2 - t - 36 = 0
Now solve for t. Note that t has 2 roots, one positive and one negative. Of course we want the positive, because time cannot be negative.
After getting the positive root of t, substitute this to equation (1) to solve for r.
Hope this helps~ :3
original speed --- x mph
time at original speed = 144/x hrs
new speed = x+4
time at new speed = 144/(x+4)
144/x - 144/(x+4) = 1/2
solve for x
let me know what you get.
To find the speed of the car, let's start by setting up an equation.
Let's assume the speed of the car is "x" mph.
According to the information provided, the car is driven 144 miles at this speed, so the time it takes to travel this distance would be:
Time = Distance / Speed
Time = 144 / x
If the car had traveled 4 mph faster, its speed would be (x + 4) mph.
According to the given information, this increase in speed would have reduced the time of the trip by 1/2 hour.
So the new time taken for the trip would be:
Time = Distance / Speed
Time = 144 / (x + 4)
The given information tells us that the new time is 1/2 hour less than the original time, so we can set up the equation:
144 / x - 144 / (x + 4) = 1/2
To solve this equation, we need to simplify and isolate the variable x.
Multiplying both sides of the equation by 2x(x + 4) to clear the fractions:
2x(x + 4)(144 / x - 144 / (x + 4)) = 2x(x + 4)(1/2)
Simplifying:
288(x + 4) - 288x = x(x + 4)
Expanding and simplifying:
288x + 1152 - 288x = x^2 + 4x
1152 = x^2 + 4x
Rearranging the equation:
x^2 + 4x - 1152 = 0
This is a quadratic equation. To solve it, we can either factor it or use the quadratic formula.
Factoring it would be quite complex, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 4, and c = -1152.
Plugging these values into the formula:
x = (-4 ± √(4^2 - 4 * 1 * -1152)) / (2 * 1)
x = (-4 ± √(16 + 4608)) / 2
x = (-4 ± √4624) / 2
The square root of 4624 is 68, so we can simplify further:
x = (-4 ± 68) / 2
Now we solve for x by using both the positive and negative values:
x = (-4 + 68) / 2 or x = (-4 - 68) / 2
x = 64 / 2 or x = -72 / 2
x = 32 or x = -36
Since speed cannot be negative in this case, we discard the negative solution.
Therefore, the speed of the car is 32 mph.