Question

Roberto rowed 20 miles downstream in 2.5 hours. The trip back, however, took him 5 hours. Find the rate that Roberto rows in still water and the rate of the current. If x is Roberto's rowing speed, and y is the speed of the river current, construct one equation modeling his downstream trip and another modeling his upstream trip using x and y .

The downstream equation is ____________

The upstream equation is ___________

Answer 1

Answer:

Roberto's speed in still water is 6 miles/hour and the river speed is 2 miles/hour

Step-by-step explanation:

Relative Speed

When a body is moving at a constant speed v, the distance traveled in a time t is:

$d=v.t$

When Roberto rows downstream, his speed in still water is added to the speed of the water, making it easier to travel the required distance.

When Roberto rows upstream, his speed in still water is affected by the speed of the water, both are subtracted and the required distance is covered in more time.

Let's call

x = Roberto's rowing speed in still water

y = Speed of the river current

The speed when rowing downstream is x+y, thus the distance traveled is

$d=(x+y).t_1$

Where t1=2.5 hours. Substituting values:

$20=(x+y)*2.5$

Rearranging, we find the downstream equation:

$2.5x+2.5y=20\qquad[1]$

The speed when rowing upstream is x-y, and the distance traveled is

$d=(x-y).t_2$

Where t2=5 hours. Substituting values:

$20=(x-y)*5$

Rearranging, we find the upstream equation:

$5x-5y=20\qquad[2]$

Multiplying [1] by 2:

$5x+5y=40$

Adding this equation to [2]:

$10x=60$

Solving:

$x=60/10=6$

Dividing [2] by 5:

$x-y=4$

Solving for y

$y=x-4=6-4=2$

Thus, Roberto's speed in still water is 6 miles/hour and the river speed is 2 miles/hour