A jogger ran 6 miles, decreased her speed by 1 mile per hour and than ran another 7 miles. If total jogging time was 2 1/42, find speed for each part of her run.
first speed --- x
second speed --- x-1
time at first speed = 6/x
time at 2nd speed = 7/(x-1)
6/x + 7/(x-1) = 2 1/42 = 85/42
times: 42x(x-1)
6(42)(x-1) + 7(42)x = 85x(x-1)
252x - 252 + 294x = 85x^2 - 85x
85x^2 - 631x + 252 = 0
x = (631 ± √312481)/170
= 7 or ...
aha , it facored
(x-7)(85x - 36) = 0
x=7 or x = 36/85
but 36/85 would make his 2nd speed negative, not very likeyly, so
x = 7
for the first leg he ran at 7 mph, for the 2nd leg he ran at 6 mph
check: 6/7 + 7/6 = (36+49)/42 = 85/42 = 2 1/42
Oh boy, this jogger sure loves a good challenge! Let's break this down and get those speeds clowning.
So, we know that the jogger ran a total distance of 6 + 7 = 13 miles. And we're given the total jogging time as 2 1/42 (or 2 hours and 1 minute).
Let's assume that the jogger's initial speed was 'x' miles per hour. Since she decreased her speed by 1 mile per hour for the second part of the run, her speed for that part would be 'x-1' miles per hour.
Now, let's dive into some math circus!
The time taken for the first 6-mile run is given by the distance divided by the speed:
6 miles / x mph = 6/x hours
The time taken for the second 7-mile run is given by:
7 miles / (x-1) mph = 7/(x-1) hours
Now, we add up these times and set it equal to 2 1/42 hours:
6/x + 7/(x-1) = 2 1/42
And now, we transform this equation into a ridiculously fun quadratic equation:
252/x(x-1) + 294/(x(x-1)) = 85/42
Time to put on our clown shoes and simplify:
252(x-1) + 294x = 85x(x-1)
Simplify further:
252x - 252 + 294x = 85x^2 - 85x
Collect like terms in the quadratic equation:
0 = 85x^2 - 631 x + 252
Oh boy, look at this goofy quadratic equation! Now, we can solve this equation using the quadratic formula, but I'll leave that part to you.
After you find the values for 'x', the initial speed, don't forget to subtract 1 mile per hour to get the speed for the second part of the run, which is 'x - 1'.
And there you have it, the clownish speeds for each part of her run!
Let's solve this step-by-step.
Let's denote the speed of the jogger for the first part of the run as "x" miles per hour.
So, the speed for the second part of the run will be "x - 1" miles per hour.
Now, let's find the time it took for the jogger to run the first part.
Distance = Speed × Time
6 miles = x miles per hour × t1 hours
Next, let's find the time it took for the jogger to run the second part.
Distance = Speed × Time
7 miles = (x - 1) miles per hour × t2 hours
We are given that the total jogging time is 2 1/42 hours.
We can convert this mixed fraction into an improper fraction.
2 1/42 = (2 × 42 + 1) / 42 = 85 / 42 = 5/2 hours
Now, we can write an equation for the total jogging time.
t1 + t2 = 5/2
We can substitute the equations for t1 and t2 in terms of x into this equation.
6 / x + 7 / (x - 1) = 5/2
Next, we can simplify and solve this equation.
12(x - 1) + 14x = 5(x)(x - 1)
12x - 12 + 14x = 5x^2 - 5x
0 = 5x^2 - 5x - 12x + 12 - 14x
0 = 5x^2 - 31x + 12
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Factoring the quadratic equation, we have:
0 = (5x - 3)(x - 4)
Setting each factor equal to zero, we get:
5x - 3 = 0 or x - 4 = 0
Solving these equations, we find:
x = 3/5 or x = 4
Since the speed cannot be negative, we discard the solution x = 3/5.
Therefore, the speed for the first part of the run is 4 miles per hour, and the speed for the second part of the run is 4 - 1 = 3 miles per hour.
To find the speeds for each part of the jogger's run, we can use the formula:
Speed = Distance / Time
Let's break down the information given:
First part of the run:
Distance = 6 miles
Speed = unknown
Time = unknown
Second part of the run:
Distance = 7 miles
Speed = unknown
Time = unknown
Total jogging time = 2 1/42 hours
We can set up two equations based on the given information to solve for the speeds of each part:
Equation 1:
6 / x = t (where x is the speed for the first part, t is the time for the first part)
Equation 2:
7 / (x - 1) = (2 1/42) - t (where x - 1 is the speed for the second part, t is the time for the first part, and (2 1/42) - t is the time for the second part)
Now, let's solve the equations step by step:
For Equation 1:
6 / x = t
t = 6 / x (Equation 1 rewritten)
For Equation 2:
7 / (x - 1) = (2 1/42) - t
7 / (x - 1) = (85/42) - t (Converting (2 1/42) to an improper fraction)
7 / (x - 1) = (85/42) - (6 / x) (Substituting t = 6 / x from Equation 1)
Now, we can solve Equation 2 for t:
7 / (x - 1) = (85/42) - (6 / x)
(7x / (x - 1)) - (6(x - 1) / (x - 1)) = 85/42 (Combining fractions)
7x - 6(x - 1) = 85/42(x - 1) (Multiplying both sides by (x - 1))
7x - 6x + 6 = (85/42)(x - 1)
x + 6 = (85/42)(x - 1) (Simplifying)
Now, we can solve Equation 2 for x:
Multiply both sides by 42:
42x + 252 = 85(x - 1)
42x + 252 = 85x - 85
42x - 85x = -85 - 252
-43x = -337
x = (-337) / (-43)
x ≈ 7.84
Now, we can substitute the value of x back into Equation 1 to find t:
t = 6 / x
t = 6 / 7.84
t ≈ 0.76
Therefore, the speed for the first part of her run is approximately 7.84 miles per hour, and the time for the first part is approximately 0.76 hours (or 45.6 minutes).
To find the speed for the second part of her run:
Speed = x - 1
Speed = 7.84 - 1
Speed ≈ 6.84 miles per hour
Hence, the speed for the second part of her run is approximately 6.84 miles per hour.
So, the jogger's speed for the first part of her run is approximately 7.84 miles per hour, while the speed for the second part is approximately 6.84 miles per hour.