During the first part of a trip, a canoeist travels 98 miles at a certain speed. The canoeist travels 10 miles on the second part of the trip at a speed 5 mph slower. The total time for the trip is 3 hours. What was the speed on the FIRST part of the trip? What was the speed on the SECOND part of the trip?

(type an integer or a decimal. Round to the nearest hundredth)

98mi @ X mi/hr.

10mi @ (x-5) mi/hr.

T = 98/x + 10/(x-5) = 3hrs.
Multiply both sides by x(x-5):
98(x-5) + 10x = 3x(x-5),
98x - 490 + 10x = 3x^2 - 15x,
-3x^2 + 108x + 15x - 490 = 0,
-3x^2 + 123x -490 = 0,
Solve using Quadratic Formula and get:
X = 36.53, and 4.47.
The required value of x is > 5.

X = 36.53 mi/hr = Speed during 1st part of trip.

X-5 = 31.53 = Speed during 2nd part of trip.

To solve this problem, we need to set up two equations based on the given information.

Let's assume the speed on the first part of the trip is represented by "x" mph.

The time taken to travel the first part of the trip can be calculated using the formula: time = distance / speed.
So, the time taken for the first part will be: 98 miles / x mph = 98/x hours.

Now, let's find the speed on the second part of the trip. We are told that the speed on the second part is 5 mph slower than the first part, so it will be (x - 5) mph.

The time taken to travel the second part of the trip is 10 miles / (x - 5) mph = 10/(x - 5) hours.

The total time for the trip is 3 hours, so the sum of the times for both parts should equal 3 hours.
Thus, the equation becomes: 98/x + 10/(x - 5) = 3.

To solve this equation, we can multiply through by the least common denominator (x(x - 5)) to eliminate the denominators:
98(x - 5) + 10x = 3x(x - 5).

Expanding and rearranging the equation gives us:
98x - 490 + 10x = 3x^2 - 15x.
3x^2 - 123x + 490 = 0.

Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Using the quadratic formula, which states: x = (-b ± sqrt(b^2 - 4ac)) / (2a), where a = 3, b = -123, and c = 490, we can find the values of x.

Calculating the values using the quadratic formula:
x = (123 ± sqrt((-123)^2 - 4 * 3 * 490)) / (2 * 3).

Simplifying the equation further:
x = (123 ± sqrt(15129 - 5880)) / 6,
x = (123 ± sqrt(9249)) / 6.

The square root of 9249 is approximately 96.15.

So, the two possible values for x are:
x = (123 + 96.15) / 6 ≈ 36.525,
x = (123 - 96.15) / 6 ≈ 4.975.

Rounding to the nearest hundredth:
The speed on the first part of the trip is approximately 36.53 mph.
The speed on the second part of the trip is approximately 4.98 mph.