A boy started on a trip across a lake by motorboat. After he had traveled 15km, the motor failed and he had to use a banca for the remaining 6 km to his destination. His average speed by motor was 4kph faster than his average speed while rowing. If the entire trip took 5 1/2 hours, what was his average speed while rowing?
Total distance = 21 km
Let x = rowing speed in kph
total trip time = 5.5 hours
15km/(x+4)kph + 6km/x kph = 5.5 hours
Multiply both sides by x(x+4), where x>0,
15x+6(x+4)=5.5x(x+4)
Solve for x and reject the negative root to get x=2 kph.
first, recall that speed, v, is distance traveled over a certain period of time:
v=d/t
thus t=d/v *we'll need this equation later*
represent the unknowns using variables:
let x = average speed of motorboat (in kph)
let y = average speed of rowing (in kph)
set-up the equations:
(1) x = y + 4
(2) (15/x) + (6/y) = 5.5 *this is t=d/v*
substitute equation (1) to (2):
15/(y+4) + 6/y = 5.5 *simplify.
15y + 6(y+4) = 5.5(y)(y+4) *multiply by 2 to make 5.5 whole number
30y + 12y + 48 = 11(y^2 + 4y)
42y + 48 = 11y^2 + 44y
11y^2 + 2y - 48=0
(11y+24)(y-2)=0
y = 2 kph *average speed of rowing
*note: the other root, y=-24/11 is extraneous since speed cannot be negative.
so there,, =)
To solve this problem, let's use the concept of average speed.
Let's assume the average speed while rowing is "x" kilometers per hour. Since the average speed by motor is 4 km/h faster than the average speed while rowing, we can say the average speed by motor is "x+4" km/h.
The boy traveled 15 km by motor and 6 km by banca, making a total distance of 15 km + 6 km = 21 km.
We know that time equals distance divided by speed.
The time spent traveling by motorboat can be calculated as:
Time = Distance / Speed
Time = 15 km / (x+4) km/h
The time spent traveling by banca (rowing) can be calculated as:
Time = Distance / Speed
Time = 6 km / x km/h
Since the total trip took 5 1/2 hours, we can write the equation:
Total Time = Time by motor + Time by banca
5.5 hours = 15 km / (x+4) km/h + 6 km / x km/h
To solve this equation, we need to find a common denominator.
Multiply the first fraction by x / x and the second fraction by (x+4) / (x+4):
5.5 hours = (15 km * x) / (x*(x+4)) + (6 km * (x+4)) / (x*(x+4))
Simplify the equation:
5.5 hours = (15x + 6x + 24) / (x^2 + 4x)
Now, cross-multiply:
5.5 hours * (x^2 + 4x) = 15x + 6x + 24
Simplify again:
5.5x^2 + 22x = 21x + 24
Move all terms to one side of the equation:
5.5x^2 + 22x - 21x - 24 = 0
Combine like terms:
5.5x^2 + x - 24 = 0
To solve this quadratic equation, we can either factor it or use the quadratic formula.
By factoring or using the quadratic formula, we can find the value of "x", which represents the average speed while rowing.