During the first part of a trip, a canoeist travels 68 miles at a certain speed. The canoeist travels 16 miles on the second part of the trip at a speed 5 mph slower. The total time for the trip is 3 hrs. What was the speed on each part of the trip?
68mi @ (X+5)mi/h.
16mi @ xmi/h.
t = 68 / (X+5) + 16 / X = 3h,
LCM = X(X+5):
(68X + 16(X+5)) / X(X+5) = 3,
(84X + 80) / X(X+5) = 3,
Multiply both sides by X(X+5):
84X + 80 = 3X(x+5),
84X + 80 = 3X^2 + 15X,
3X^2 + 15X - 84X - 80 = 0,
3X^2 - 69X - 80 = 0.
Solve for X using the Quad. Formula and
get:
X = 24.1062, and - 1.1062.
Select the + value of X:
X = 24.1062mi/h.
X+5 = 29.1062mi/h.
during the first part of the trip a canoeist travels 24 miles at a certain speed. the canoeist travels 7 miles on the second part of the trip at speed of 5 mpg slower. the total time for the trip is 3 hrs. what is the speed of each trip
To solve this problem, let's break it down into two parts: the first part of the trip and the second part of the trip.
Let's assume that the speed of the canoeist during the first part of the trip is "x" mph.
Therefore, the time taken for the first part of the trip can be calculated using the formula time = distance / speed:
time taken for the first part = 68 miles / x mph
Now, let's find the speed of the canoeist during the second part of the trip. We know that the speed during the second part is 5 mph slower than during the first part, so the speed would be x-5 mph.
The time taken for the second part of the trip can be calculated in a similar way:
time taken for the second part = 16 miles / (x - 5) mph
According to the problem, the total time for the trip is 3 hours. So, we can write the equation:
time taken for the first part + time taken for the second part = total time
68 / x + 16 / (x - 5) = 3
Now, we need to solve this equation to find the value of "x" (the speed during the first part of the trip).
To simplify the equation, we can first multiply both sides by x(x - 5) to eliminate the fractions:
68(x - 5) + 16x = 3x(x - 5)
68x - 340 + 16x = 3x^2 - 15x
0 = 3x^2 - 15x - 84x + 340
Rearranging the terms and combining like terms:
0 = 3x^2 - 99x + 340
Now, we need to factorize this quadratic equation to find the value(s) of "x".
Factoring this equation can be a bit challenging, so let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation: a = 3, b = -99, and c = 340.
Calculating the values:
x = [-(-99) ± sqrt((-99)^2 - 4 * 3 * 340)] / (2 * 3)
x = (99 ± sqrt(9801 - 4080)) / 6
x = (99 ± sqrt(5721)) / 6
Since the discriminant (b^2 - 4ac) is positive, we have real and distinct solutions.
Calculating further:
x = (99 ± 75.64) / 6
x1 = (99 + 75.64) / 6 = 174.64 / 6 ≈ 29.11 mph
x2 = (99 - 75.64) / 6 = 23.36 / 6 ≈ 3.89 mph
Therefore, the speed on each part of the trip is approximately 29.11 mph during the first part and 3.89 mph during the second part.