A boy can row a boat at a constant rate of 5 mi/hr in still water, as indicated in the figure. He rows upstream for 24 minutes and then rows downstream, returning to his starting point in another 16 minutes. (Round your answers to two decimal places.)
(A) Find rate of the current
(B) Find travel distance traveled
rate of current --- x mph
let the distance each way by y miles
so upstream distance:
y = (24/60)(5-x)
downstream distance
y = (16/60)(5+x)
but they are both equal to y, so
24(5-x)/60 = 16(5+x)/60
times 60
24(5-x) = 16(5+x)
120 - 24x = 80 + 16x
-40x = -40
x = 1
So the current is 1 mph
y = 16(6)/60 = 1.6 miles
(as a check sub in x = 1 into the other y = ... equation)
Two factory plants are making TV panels. Yesterday, Plant A produced
5000
fewer panels than Plant B did. Three percent of the panels from Plant A and
2%
of the panels from Plant B were defective. How many panels did Plant B produce, if the two plants together produced
750
defective panels?
To find the rate of the current and the travel distance, we can use the formula:
Boat speed in still water = (Speed downstream + Speed upstream) / 2
(A) Let's find the rate of the current:
Given:
Boat speed in still water is 5 mi/hr.
Time taken to row upstream is 24 minutes, which is 24/60 = 0.4 hours.
Time taken to row downstream is 16 minutes, which is 16/60 = 0.27 hours.
Let's assume the rate of the current is x mi/hr.
Boat speed downstream = 5 + x mi/hr
Boat speed upstream = 5 - x mi/hr
Using the formula mentioned above:
5 = (5 + x + 5 - x) / 2
10 = 10 / 2
10 = 5
The rate of the current is 0 mi/hr.
(B) Now let's find the travel distance:
To find the travel distance, we will use the formula: Distance = Speed x Time.
For the upstream journey:
Distance upstream = Speed upstream x Time upstream
= (5 - 0) mi/hr x 0.4 hr
= 2 mi
For the downstream journey:
Distance downstream = Speed downstream x Time downstream
= (5 + 0) mi/hr x 0.27 hr
= 1.35 mi
Total travel distance = Distance upstream + Distance downstream
= 2 mi + 1.35 mi
= 3.35 mi
Therefore, the travel distance is 3.35 miles.
To find the rate of the current, we can use the concept of relative velocity. When the boy rows upstream, he is rowing against the current, so his effective speed will be reduced by the speed of the current. When he rows downstream, he will be aided by the current, so his effective speed will be increased by the speed of the current.
Let's assume the speed of the current is "c" (mi/hr). Since the boy can row at a constant rate of 5 mi/hr in still water, his effective speed when rowing upstream will be 5 - c mi/hr, and his effective speed when rowing downstream will be 5 + c mi/hr.
Given that the boy takes 24 minutes (which is 24/60 = 0.4 hours) to row upstream and another 16 minutes (which is 16/60 = 0.27 hours) to row downstream, we can set up the following equations:
Distance upstream = Speed upstream * Time upstream
Distance downstream = Speed downstream * Time downstream
Using the formula Distance = Speed * Time, we can rewrite the equations as:
Distance upstream = (5 - c) * 0.4
Distance downstream = (5 + c) * 0.27
Rearranging the equations, we get:
(5 - c) * 0.4 = Distance upstream
(5 + c) * 0.27 = Distance downstream
Since the distances upstream and downstream are equal (as the boy returns to his starting point), we can set the equations equal to each other:
(5 - c) * 0.4 = (5 + c) * 0.27
Now, we can solve the equation for "c" to find the rate of the current.