During the first part of a trip, a canoeist travels 93 miles at a certain speed. The canoeist travels 19 miles on the second part of the trip at a speed of 5 mph slower. The total time for the trip is 3 hrs. What was the speed on each part?
let the "certain" speed be x mph
so time at the certain speed = 93/x
slower speed = x-5
time at slower speed = 19/(x-5)
so 93/x + 19/(x-5) = 3
times x(x-5)
93(x-5) + 19x = 3x(x-5)
93x - 465 + 19x = 3x^2 - 15x
3x^2 - 127x + 465 = 0
x = (127 ± √(10549)/6 = 38.28 or 4.045 mph
the "certain" speed of 4.045 mph would give us a negative speed for the second leg, so we have to reject that.
so that leaves us with a speed of 38.28 mph for the first part, and 33.28 mph for the second part of the trip.
(notice that 93/38.28 + 19/33.28 = 3)
Even though the answer fits the given data, the question is totally ridiculous.
I am an avid canoist and to be able to canoe at 38 mph is totally absurd.
Thank you so much for your help, and I would be interested in watching someone try to paddle a canoe or any boat that fast. Bet their arms would be very tired before they went 1/4 mile.
To find the speed on each part of the trip, we can use the formula:
Speed = Distance / Time
Let's assign variables for the unknowns. Let's call the speed on the first part of the trip "x" mph, and the speed on the second part "x - 5" mph.
Now, let's calculate the time for each part of the trip:
Time for the first part = Distance / Speed = 93 miles / x mph
Time for the second part = Distance / Speed = 19 miles / (x - 5) mph
According to the problem, the total time for the trip is 3 hours:
Time for the first part + Time for the second part = 3 hours
93 / x + 19 / (x - 5) = 3
To solve this equation, let's multiply through by x(x - 5) to clear the denominators:
93(x - 5) + 19x = 3x(x - 5)
Distribute and simplify:
93x - 465 + 19x = 3x^2 - 15x
Rearrange and combine like terms:
3x^2 - 15x - 112x + 465 - 19x = 0
3x^2 - 146x + 465 = 0
Now, we can solve this quadratic equation for x using factoring, completing the square, or the quadratic formula. In this case, let's use factoring:
(x - 15)(3x - 31) = 0
Setting each factor equal to zero:
x - 15 = 0 --> x = 15
3x - 31 = 0 --> x = 10.33 (rounded to two decimal places)
Since speed cannot be negative, we discard the negative value. Therefore, x = 15 mph.
The speed on the first part of the trip is 15 mph, and the speed on the second part is 15 - 5 = 10 mph.