during the first part of a trip, a canoeist travels 47 miles at a certain speed. the canoeist travels 16miles on the second part of the trip at a speed 5mph slower. the total time for the trip is 4 hours. what was the speed on each part of the trip.

To solve this problem, we can use a system of equations.

Let's denote the speed during the first part of the trip as "x" mph. The speed during the second part of the trip would then be "x - 5" mph (since the speed is 5 mph slower).

We know the distance covered during the first part of the trip is 47 miles, and during the second part, it is 16 miles. The total time for the trip is 4 hours.

Now, we can use the formula "distance = speed × time" to create the following equations:

Equation 1: 47 = x * t1
Equation 2: 16 = (x - 5) * t2

We also know that the total time is 4 hours, so we can write:

t1 + t2 = 4

Now, let's solve this system of equations.

From Equation 1, we can express t1 in terms of x:

t1 = 47 / x

From Equation 2, we can express t2 in terms of x:

t2 = 16 / (x - 5)

Substituting these values into the equation t1 + t2 = 4, we get:

47 / x + 16 / (x - 5) = 4

To simplify this equation, we can multiply both sides by (x * (x - 5)) to eliminate the denominators:

47(x - 5) + 16x = 4x(x - 5)

Simplifying further:

47x - 235 + 16x = 4x^2 - 20x

Rearranging the equation:

4x^2 - 20x - 63x + 235 - 16x = 0

Combining like terms:

4x^2 - 99x + 235 = 0

Unfortunately, this equation doesn't simplify easily or factor nicely. To accurately determine the values of x, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 4, b = -99, and c = 235.

Using the quadratic formula to solve for x, we get:

x = (-(-99) ± √((-99)^2 - 4 * 4 * 235)) / (2 * 4)

Simplifying:

x = (99 ± √(9801 - 3760)) / 8

x = (99 ± √6041) / 8

Therefore, the speed during the first part of the trip is approximately:

x = (99 + √6041) / 8 ≈ 13 mph

And the speed during the second part of the trip is approximately:

x - 5 = 13 - 5 = 8 mph

So, the speed during the first part of the trip is approximately 13 mph, and during the second part, it is approximately 8 mph.