Ed is planning daily walking workouts on the same distance of 12 km in the morning and in the evening. Usually he is walking at constant rate, however he planned his rate to be 1 km/h more in the morning than in the evening. Given he is willing to spend 5 hours and 24 minutes daily on these workouts, what should his morning rate be, in km/h?
5 km/h
r = X km/h in the mornings.
r = x-1 km/h in the evening.
t1 + t2 = 5 .4 h.
x * t1 = 12.
t1 = 12/x.
(x-1) * t2 = 12.
t2 = 12/(x-1).
t1 + t2 = 5.4.
12/x + 12/(x-1) = 5.4.
12(x-1) + 12x = 5.4x(x-1),
12x - 12 +12x = 5.4x^2 - 5.4x,
5.4x^2 - 24x - 5.4x + 12 = 0.
5.4x^2 - 29.4x + 12 = 0,
sqrt(B^2 - 4AC) = 24.6.
X = 29.4/10.8 +- 24.6/10.8 = 2.72 +- 2.28 = 5 km/h, and 0.44.
X = 5 km/h in morning. You are correct!!.
To find Ed's morning rate in km/h, we can set up an equation using the given information.
Let's assume that Ed's rate in the evening is "x" km/h. As per the question, his rate in the morning will be 1 km/h more than in the evening, so his morning rate would be "x + 1" km/h.
Since Ed walks the same distance of 12 km in the morning and evening, we can calculate the time taken for each workout.
The time taken for the morning workout = distance/rate = 12 km/(x + 1) km/h
The time taken for the evening workout = distance/rate = 12 km/x km/h
According to the question, Ed is willing to spend a total of 5 hours and 24 minutes (which is equivalent to 5.4 hours) on these workouts. Therefore, the sum of the time taken for both workouts should be equal to 5.4 hours.
(12 km/(x + 1) km/h) + (12 km/x km/h) = 5.4 hours
To solve this equation for "x" (the rate in the evening), we can multiply both sides of the equation by "x(x + 1)", which will eliminate the denominators:
12x + 12(x + 1) = 5.4x(x + 1)
Now, simplify and solve the equation for "x":
12x + 12x + 12 = 5.4x^2 + 5.4x
24x + 12 = 5.4x^2 + 5.4x
Rearrange the equation to bring all terms to one side:
5.4x^2 + 5.4x - 24x - 12 = 0
5.4x^2 - 18.6x - 12 = 0
Next, we can solve this quadratic equation using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 5.4, b = -18.6, and c = -12.
x = (-(-18.6) ± sqrt((-18.6)^2 - 4 * 5.4 * (-12))) / (2 * 5.4)
Simplifying further:
x = (18.6 ± sqrt(345.96 + 259.2)) / 10.8
x = (18.6 ± sqrt(605.16)) / 10.8
Now, we have two possible values for x:
x = (18.6 + sqrt(605.16)) / 10.8
x ≈ 4.1906
x = (18.6 - sqrt(605.16)) / 10.8
x ≈ -1.1906
Since rate (speed) cannot be negative, we discard the negative solution.
Therefore, the rate in the evening (x) is approximately 4.1906 km/h.
Finally, as Ed's morning rate is 1 km/h more than his evening rate, we can calculate his morning rate:
Morning rate (x + 1) = 4.1906 + 1 = 5.1906 km/h
Hence, Ed's morning rate should be approximately 5.1906 km/h.