Solving a Real-World Application of Speed and Current with Algebra
Ever faced a problem where you had to calculate the speed of a boat in still water when you know the distance it covers downstream and upstream, along with the time and speed of the current? This article guides you through a real-world application of speed and current using algebra, helping you understand how to solve such problems step by step.
Problem Statement
A boat goes 28 km downstream and while returning covers only 75% of the distance covered downstream. If the boat takes 3 hours more to cover the upstream distance than the downstream distance, what is the speed of the boat in still water if the speed of the current is 5/9 meters per second?
Step-by-Step Solution
Step 1: Define Variables
Let the speed of the boat in still water be b km/h. The speed of the current is given as c 5/9 m/s. The distance covered downstream is d 28 km. The distance covered upstream is 0.75 * 28 21 km (75% of 28 km).Step 2: Convert Current Speed to Consistent Units
To ensure our calculations are consistent, we convert the speed of the current from meters per second (m/s) to kilometers per hour (km/h).
c (5/9) * (3600/1000) 20 km/h
Step 3: Determine Effective Speeds
Downstream speed: b c b 20 km/h Upstream speed: b - c b - 20 km/hStep 4: Calculate Time Taken for Journeys
Time taken downstream: t_d 28 / (b 20) Time taken upstream: t_u 21 / (b - 20)Step 5: Set Up the Time Difference Equation
Given that the boat takes 3 hours more to cover the upstream distance than the downstream distance, we can write:
t_u t_d 3
Substituting the expressions for t_u and t_d:
21 / (b - 20) 28 / (b 20) 3
Step 6: Solve the Equation
First, eliminate the denominators by multiplying both sides by (b - 20)(b 20):
21(b 20) 28(b - 20) 3(b - 20)(b 20)
Expanding and rearranging terms:
21b 420 28b - 560 3b^2 - 400
0 3b^2 - 7b - 2180
Step 7: Quadratic Equation and Solution
We now have a standard quadratic equation in the form:
3b^2 - 7b - 2180 0
To solve this, we use the quadratic formula:
b (-B ± sqrt(B^2 - 4AC)) / (2A)
Where A 3, B -7, C -2180.
First, calculate the discriminant:
D B^2 - 4AC (-7)^2 - 4 * 3 * (-2180) 49 26160 26109
Now, substitute the values into the quadratic formula:
b (-(-7) ± sqrt(26109)) / (2 * 3) (7 ± 161.5) / 6
Calculating the two possible values for b:
b (154.5) / 6 25.75 km/h (valid speed since it is positive)
b (-168.5) / 6 (not valid as speed cannot be negative)
Thus, the speed of the boat in still water is approximately: boxed{25.75 km/h}