dURING THE FIRST PART OF A TRIP, A CONOEIST TRAVELS53 MILES AT A CERTAIN SPEED . The canoeist travels 9 miles on the second part of the tripat a speed 5 mph lower.The total time for the trip is 5 hrs, what was the speed on each part of the trip. The first part of the trip was ..... mph. (simplify your answer.type an integar or decimal.round to nearest hundreth).
Speed (S) = Distance (D)/Time (T)
Therefore T = D/S
5 = 53/S + 9/(S-5)
Solve for S.
I hope this helps. Thanks for asking.
To solve this problem, we'll need to use the formula:
Speed = Distance / Time
Let's start by calculating the time it took for the canoeist to travel the first part of the trip. We know that the distance covered in the first part is 53 miles, and we'll assume the speed is x mph.
Time for the first part = Distance / Speed = 53 / x
Next, let's calculate the time it took for the second part of the trip. We know that the distance covered in the second part is 9 miles and the speed is 5 mph lower than the first part's speed, which is (x - 5) mph.
Time for the second part = Distance / Speed = 9 / (x - 5)
According to the problem, the total time for the trip is 5 hours. So we can set up the equation:
Time for the first part + Time for the second part = Total time
53 / x + 9 / (x - 5) = 5
To solve this equation, we need to get rid of the denominators by multiplying through by the least common denominator (LCD), which is x(x - 5). This will give us a quadratic equation.
Multiplying every term by x(x - 5):
53(x - 5) + 9x = 5x(x - 5)
Simplifying and expanding:
53x - 265 + 9x = 5x^2 - 25x
Combining like terms:
62x - 265 = 5x^2 - 25x
Rearranging the equation to bring all terms to one side:
5x^2 - 87x + 265 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In our case, a = 5, b = -87, and c = 265. Plugging these values into the formula, we get:
x = (-(-87) ± sqrt((-87)^2 - 4 * 5 * 265)) / (2 * 5)
x = (87 ± sqrt(7569 - 5300)) / 10
x = (87 ± sqrt(2269)) / 10
Using a calculator or an online square root calculator, we find that sqrt(2269) ≈ 47.68.
Now, plugging this value into the equation:
x = (87 ± 47.68) / 10
Solving for both possibilities:
x1 = (87 + 47.68) / 10 = 13.37 mph
x2 = (87 - 47.68) / 10 = 3.32 mph
Therefore, the speed for the first part of the trip was approximately 13.37 mph.