A boat travels down the river in 2.6 hours. The return trip up river takes 3.9 hours. A one way trip down or up river is a distance of 21 miles. If the boat is traveling at maximum speed, what is the boats speed measured in waters with no current? You need to use systems to solve this problem. Round your speed to the tenth of a mile per hour. You do not need to include units as the problem requires a value in miles per hour.

Question

A boat travels down the river in 2.6 hours. The return trip up river takes 3.9 hours. A one way trip down or up river is a distance of 21 miles. If the boat is traveling at maximum speed, what is the boats speed measured in waters with no current? You need to use systems to solve this problem. Round your speed to the tenth of a mile per hour. You do not need to include units as the problem requires a value in miles per hour.

Answer 1

since distance = time*speed, if the boat=b and the river=r mph,

2.6(b+r) = 21
3.9(b-r) = 21
b = 175/26 = 6.73 mph
r = 25/36 = 1.53 mph

Answer 2

To solve this problem using systems, we'll create two equations based on the information given.

Let's assume the speed of the boat in still water as 'x' miles per hour and the speed of the current as 'y' miles per hour.

Equation 1: Speed of boat downstream = Speed of boat in still water + Speed of current
Equation 2: Speed of boat upstream = Speed of boat in still water - Speed of current

We are given the following information:
- The boat travels down the river in 2.6 hours, which is the time it takes to cover a distance of 21 miles.
- The return trip upriver takes 3.9 hours, which is also the time it takes to cover a distance of 21 miles.

Using the formula distance = speed * time:
Equation 3: 21 = (x + y) * 2.6 (downstream)
Equation 4: 21 = (x - y) * 3.9 (upstream)

We can now solve these equations simultaneously to find the values of 'x' (speed of the boat in still water) and 'y' (speed of the current).

Let's simplify Equation 3:
21 = 2.6x + 2.6y (Divided both sides by 2.6)

Now, let's simplify Equation 4:
21 = 3.9x - 3.9y (Divided both sides by 3.9)

Now we have a system of equations:

Equation 5: 2.6x + 2.6y = 21
Equation 6: 3.9x - 3.9y = 21

We can solve this system by elimination or substitution method.

Let's use the elimination method. Multiply both sides of Equation 5 by 1.5 (to make the coefficients of 'y' in both equations the same):

Equation 7: 3.9x + 3.9y = 31.5

Now, add Equation 6 and Equation 7:
(3.9x - 3.9y) + (3.9x + 3.9y) = 21 + 31.5

Simplifying the equation: 7.8x = 52.5

Divide both sides by 7.8:
x = 52.5 / 7.8 ≈ 6.73

The speed of the boat in still water is approximately 6.73 miles per hour.

We do not need to use Equation 4 to find the speed of the boat, as it is not specified in the problem. We need Equation 3 only to find 'x'.

Hence, the boat's speed in still water is approximately 6.7 miles per hour (rounded to the nearest tenth).