Julie and Eric row their boat (at a constant speed) 21 miles downstream for 3 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 7 hours. Find the rate of the current.
Their speed downstream = 21/3 = 7mph
Their speed upstream = 21/7 = 3mph
So, the river speed is (7-3)/2 = 2mph
Their rowing speed is 5mph.
To find the rate of the current, we need to set up a system of equations using the information given.
Let's assume the speed of the boat in still water is "B" and the speed of the current is "C".
When Julie and Eric row downstream with the current, their effective speed will be the sum of the boat speed and the current speed:
Speed downstream = B + C
The distance traveled downstream is 21 miles, and they took 3 hours to cover that distance:
Distance = Speed × Time
21 = (B + C) × 3
When Julie and Eric row upstream against the current, the effective speed will be the difference between the boat speed and the current speed:
Speed upstream = B - C
The distance traveled upstream is still 21 miles, but it took them 7 hours to cover that distance:
21 = (B - C) × 7
Now we have a system of two equations:
1) 21 = (B + C) × 3
2) 21 = (B - C) × 7
To solve this system, we can use the method of substitution or elimination.
Let's solve using substitution:
From equation 1, we have B + C = 21/3, which simplifies to B + C = 7.
Rearranging equation 2, we have 21/7 = B - C, which simplifies to B - C = 3.
Now we have a new simplified system:
1) B + C = 7
2) B - C = 3
To solve the system, we can add equations 1 and 2:
(B + C) + (B - C) = 7 + 3
2B = 10
B = 10/2
B = 5
Now, substitute the value of B back into one of the equations:
5 + C = 7
C = 7 - 5
C = 2
Therefore, the rate of the current is 2 miles per hour.
To find the rate of the current, we need to understand the concept of relative speed. Let's break down the problem into steps:
Step 1: Define the variables.
- Let r be the rate of the boat (in still water), measured in miles per hour.
- Let c be the rate of the current, measured in miles per hour.
Step 2: Determine the downstream speed.
The downstream speed is the rate of the boat plus the rate of the current, since they are both helping the boat move forward. Therefore, the speed downstream is (r + c) miles per hour.
We can calculate the total distance traveled downstream using the formula:
Distance = Speed × Time.
In this case, the distance is given as 21 miles and the time is given as 3 hours. So, we can write the equation as:
21 = (r + c) × 3.
Step 3: Determine the upstream speed.
The upstream speed is the rate of the boat minus the rate of the current, since they are opposing each other. Therefore, the speed upstream is (r - c) miles per hour.
We can calculate the total distance traveled upstream using the same formula:
Distance = Speed × Time.
In this case, the distance is also 21 miles, but the time is given as 7 hours. So, we can write the equation as:
21 = (r - c) × 7.
Step 4: Solve the equations.
We now have a system of two equations with two variables. We can solve these equations simultaneously to find the values of r and c.
From equation (1): 21 = (r + c) × 3,
we can divide both sides by 3 to get:
7 = r + c.
From equation (2): 21 = (r - c) × 7,
we can divide both sides by 7 to get:
3 = r - c.
Step 5: Solve for the current rate.
To solve for c, we can subtract equation (2) from equation (1) since the c terms will cancel out:
7 - 3 = (r + c) - (r - c).
4 = 2c.
Divide both sides by 2:
2 = c.
Therefore, the rate of the current is 2 miles per hour.