lori recently rode her bike to her friend's house, 8 miles away. On her way back she rode 15mph faster than on her way there. She spent a total of 2 hours biking. What was her average speed each way?
since time = distance/speed,
8/s + 8/(s+15) = 2
So the speeds were 5 and 20 mi/hr
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To find Lori's average speed each way, we need to set up a system of equations:
Let x be the speed (in mph) Lori rode to her friend's house.
Since she rode 15mph faster on her way back, her speed on the way back would be x + 15 mph.
We know that the distance to her friend’s house is 8 miles.
We also know that the total time spent biking is 2 hours.
Using the formula: Speed = Distance / Time, we can set up the following equations:
On the way there: Speed = 8 / x
On the way back: Speed = 8 / (x + 15)
She spent a total of 2 hours biking, so we can set up the equation:
Time + Time = 2
8 / x + 8 / (x + 15) = 2
To solve this equation, we can multiply all terms by x(x+15) to eliminate the denominators:
(8(x+15)) + (8x) = 2x(x+15)
8x + 120 + 8x = 2x^2 + 30x
Combining like terms:
16x + 120 = 2x^2 + 30x
Rearranging the equation to set it equal to zero:
2x^2 + 14x - 120 = 0
Next, we can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 14, and c = -120. Substituting these values into the quadratic formula:
x = (-14 ± √(14^2 - 4(2)(-120))) / (2(2))
x = (-14 ± √(196 + 960)) / 4
x = (-14 ± √1156) / 4
x = (-14 ± 34) / 4
Now, we have two possible values for x:
x1 = (-14 + 34) / 4 = 20 / 4 = 5
x2 = (-14 - 34) / 4 = -48 / 4 = -12
Since speed can't be negative, we discard the negative value. Therefore, Lori's speed on her way to her friend's house was 5 mph.
To find her speed on the way back, we can substitute this value back into the equation for the speed on the way back:
Speed = 5 + 15 = 20 mph
Therefore, Lori's average speed each way was 5 mph on the way there and 20 mph on the way back.
To find Lori's average speed each way, we need to determine her speed on the way there and her speed on the way back.
Let's assume Lori's speed on the way to her friend's house is "x" mph.
According to the problem, on her way back, she rode 15 mph faster than on her way there. Therefore, her speed on the way back would be "x + 15" mph.
Given that the distance to her friend's house is 8 miles, we can use the formula: Speed = Distance / Time to find the time it takes for each trip.
On her way there:
Time = Distance / Speed
Time = 8 miles / x mph
Time = 8/x hours
On her way back:
Time = Distance / Speed
Time = 8 miles / (x + 15) mph
Time = 8/(x + 15) hours
The total time Lori spent biking is given as 2 hours. Therefore, the sum of the individual times should be equal to 2 hours:
8/x + 8/(x + 15) = 2
To solve this equation, we can multiply through by x(x + 15) to eliminate the denominators:
8(x + 15) + 8x = 2x(x + 15)
Expanding the equation and simplifying, we get:
8x + 120 + 8x = 2x^2 + 30x
Rearranging terms to form a quadratic equation:
0 = 2x^2 + 30x - 16x - 120
Simplifying the equation further:
0 = 2x^2 + 14x - 120
Next, we can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's solve it using factoring:
0 = (2x - 10)(x + 12)
Setting each factor equal to zero and solving for x:
2x - 10 = 0 --> 2x = 10 --> x = 5 (ignoring the -12 value as it is not relevant for this scenario)
So, Lori's speed on the way to her friend's house is 5 mph. Since she rode 15 mph faster on the way back, her speed on the way back would be 5 + 15 = 20 mph.
Therefore, Lori's average speed each way is:
Average speed on the way there = 5 mph
Average speed on the way back = 20 mph