Problem 4
c = speed of current
p = paddling speed in still water
Against current:
speed = p-c (start with paddling speed and subtract off current's speed)
speed = 2 (given)
p-c = 2
p = 2+c
With the current
speed = p+c (now the current is speeding things up, so we add on c)
speed = 3 (given)
p+c = 3
2+c+c = 3 ... plug in p = 2+c; solve for c
2+2c = 3
2+2c-2 = 3-2
2c = 1
c = 1/2
c = 0.5
Since c = 0.5, we can use this to find p
p = 2+c
p = 2+0.5
p = 2.5
So,
The speed of the river current is 0.5 mph
Rita's paddling speed in still water is 2.5 mph
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Problem 5
Terms to know:
headwind = wind that slows the plane down (wind is coming from the head of the plane flowing in the opposite direction of the plane's intended direction)
tailwind = wind that speeds the plane up (wind is coming from the tail of the plane flowing in the same direction of the plane's intended direction)
d = distance = 255 miles
p = speed of plane in still air
w = speed of wind
Against the wind (headwind), the plane travels 1.7 hours at some speed p-w. We start with the plane's speed in still air (p) and subtract off the wind speed because the wind is slowing the plane down. So the first equation is
(p-w)*1.7 = 255
since the plane travels 255 miles
I'm using the formula d = r*t
d = distance
r = rate or speed
t = time
Divide both sides of (p-w)*1.7 = 255 by 1.7 and we get p-w = 150
Then add w to both sides and we have p = 150+w
Similarly, the second equation is
(p+w)*1.5 = 255
since the tailwind speeds the plane up from p to p+w, the time is 1.5 hrs and the distance is the same (255 mi)
Plug the equation p = 150+w into the second equation
(p+w)*1.5 = 255
(150+w+w)*1.5 = 255
(150+2w)*1.5 = 255
150*1.5+2w*1.5 = 255
225+3w = 255
225+3w-225 = 255-225
3w = 30
3w/3 = 30/3
w = 10
Use w = 10 to find p
p = 150+w
p = 150+10
p = 160
So,
wind speed = 10 mph
speed of plane in still air = 160 mph