Kiran drove from City A to City B, a distance of 288 mi. She increased her speed by 19 mi/h for the 400-mi trip from City B to City C. If the total trip took 8 h, what was her speed from City A to City B?
Let's call the speed at which Kiran drove from City A to City B as "x" mi/h.
We know that the distance from City A to City B is 288 mi, so the time taken for this part of the trip can be calculated using the formula: time = distance / speed.
Therefore, the time taken for Kiran to drive from City A to City B is 288 / x hours.
Next, we are given that for the 400-mi trip from City B to City C, Kiran increased her speed by 19 mi/h. So her speed for this part of the trip is (x + 19) mi/h.
Using the same formula as before, the time taken for Kiran to drive from City B to City C is 400 / (x + 19) hours.
We are also given that the total trip took 8 hours. Therefore, the sum of the times taken for both parts of the trip should equal 8 hours:
288 / x + 400 / (x + 19) = 8
To solve this equation for "x," we can find a common denominator and simplify:
288(x + 19) + 400x = 8x(x + 19)
288x + 5472 + 400x = 8x^2 + 152x
688x + 5472 = 8x^2 + 152x
Rearranging the equation:
8x^2 - 536x - 5472 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. For simplicity, we will use the quadratic formula:
x = [-(-536) ± √((-536)^2 - 4(8)(-5472))] / (2 * 8)
Simplifying further:
x = (536 ± √(286,096 - 175,104)) / 16
x = (536 ± √110,992) / 16
x = (536 ± 332.88) / 16
Now we can consider both possible values for "x":
For the positive value:
x = (536 + 332.88) / 16
x ≈ 42.99 mi/h
For the negative value:
x = (536 - 332.88) / 16
x ≈ 14.05 mi/h
Since speed cannot be negative, we can conclude that Kiran's speed from City A to City B was approximately 42.99 mi/h.
To find the speed from City A to City B, we first need to consider the time it took to travel between the cities.
Let's assume the speed from City A to City B is represented by x mi/h. We can use the formula: Time = Distance / Speed
For the first leg of the trip from City A to City B:
Time1 = 288 mi / x mi/h
For the second leg of the trip from City B to City C:
Time2 = 400 mi / (x + 19) mi/h
Given that the total trip took 8 hours, we can write the equation: Time1 + Time2 = 8 hrs
Substituting the values, we have:
288 / x + 400 / (x + 19) = 8
To solve this equation, we can multiply through by (x + 19) to eliminate the fractions:
288(x + 19) + 400x = 8(x * (x + 19))
288x + 5472 + 400x = 8(x^2 + 19x)
Rearranging and simplifying the equation:
8x^2 + 240x - 5472 = 0
This is a quadratic equation that we can solve using factoring, completing the square, or the quadratic formula.
By factoring, we get:
(4x - 104)(2x + 52) = 0
So we have two potential solutions:
1) 4x - 104 = 0 -> 4x = 104 -> x = 26
2) 2x + 52 = 0 -> 2x = -52 -> x = -26
Since speed cannot be negative, we discard the second solution.
Therefore, Kiran's speed from City A to City B was 26 mi/h.
time = distance / rate
(288 / s) + [400 / (s + 19)] = 8
288 s + (19 * 288) + 400 s = 8 * s * (s + 19)
36s + 684 + 50 s = s^2 + 19 s
0 = s^2 - 67 s - 684 ... solve the quadratic for s
Kiran has a heavy foot...