During the first part of a trip a canoeist travels 55 miles at a certain speed the canoeist travels 15miles on the second part of the trip, 5 miles per hour slower. The total time for the trip is 4hours. What was the speed on each part of the trip.
This is what I have so far, then I am lost
t1= 55/r t2=15/r-5 =4
r (r-5) (55/r + 15/r-5) =4
55 (r-5) +15r=4r (r-5)
55r-275 + 15r=4r^2-20r
Now I am lost.
You are practically there!
Combine terms and rearrange to get into the form:
4r^2 -90r + 275 = 0
Use the quadratic formula to find r. There will be two answers. One of the answers is the result if the canoeist reverses direction (r-5 is negative).
r-5 being negative results in a negative t2 and so should be ignored.
To solve this problem, let's break it down step by step:
1. Let's denote the speed during the first part of the trip as r mph.
2. We know that the distance covered during the first part of the trip is 55 miles. Therefore, the time taken is 55/r hours.
3. We also know that the speed during the second part of the trip is 5 mph slower than the first part, which means the speed is (r-5) mph.
4. The distance covered during the second part of the trip is 15 miles. Therefore, the time taken is 15/(r-5) hours.
5. The total time for the trip is given as 4 hours.
6. Therefore, the equation we can form is: 55/r + 15/(r-5) = 4.
Now, let's solve the equation:
1. Multiply both sides of the equation by r(r-5) to eliminate the denominators: 55(r-5) + 15r = 4r(r-5).
2. Expand the equation: 55r - 275 + 15r = 4r^2 - 20r.
3. Combine like terms: 70r - 275 = 4r^2 - 20r.
4. Rearrange the equation to put it in standard quadratic form: 4r^2 - 90r + 275 = 0.
5. Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. However, this quadratic equation cannot be factored easily, so we will use the quadratic formula:
r = (-b ± √(b^2 - 4ac))/(2a).
In our equation, a = 4, b = -90, and c = 275.
6. Plugging these values into the quadratic formula, we get: r = (-(-90) ± √((-90)^2 - 4*4*275))/(2*4).
7. Simplifying further: r = (90 ± √(8100 - 4400))/8.
8. Continuing to simplify: r = (90 ± √3700)/8.
9. Taking the square root of 3700, we get: r ≈ (90 ± 60.83)/8.
Now, we have two possible values for r:
a) r ≈ (90 + 60.83)/8 ≈ 150.83/8 ≈ 18.85 mph.
b) r ≈ (90 - 60.83)/8 ≈ 29.17/8 ≈ 3.65 mph (but this value doesn't make sense as it implies negative speed).
Therefore, the speed during the first part of the trip is approximately 18.85 mph, and the speed during the second part of the trip is approximately (18.85 - 5) = 13.85 mph.