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river jet boats
Algebra 2 help me out please ?
1 A "Local" train leaves a station and runs at an average rate of 35 mph. An hour and a half later an "Express" train leaves the station and travels at an average rate of 56 mph on a parallel track. How many hours after it starts will the Express overtake the Local?
2 With a certain tail wind a jet aircraft arrives at its destination, 1,890 miles away, in 3 hours. Flying against the same wind, the plane makes the return trip in 3 3/8 hours. Find the wind speed and the plane's airspeed.
3 The excursion boat on the river takes 2½ hours to make the trip to a point 12 miles upstream and to return. If the rate at which the boat travels in still water is 5 times the rate of the river current, what is the rate of the current?
1 A "Local" train leaves a station and runs at an average rate of 35 mph. An hour and a half later an "Express" train leaves the station and travels at an average rate of 56 mph on a parallel track. How many hours after it starts will the Express overtake the Local?
Variables:
d = distance
r = rate
t = time
HINT: Write what you know in a table.
......................... distance ....... rate ...... time
local train …...... d ...............… 35 ....… t + 1.5
express train .... d ................… 56 ....… t
Remember that the distance formula is d = r * t. Apply this to the table.
......................... distance .............. rate ...... time
local train …...... d = 35(t + 1.5) ...… 35 ....… t + 1.5
express train .... d = 56t...............… 56 ....… t
You have 2 equations.
d_local = 35(t + 1.5)
d_express = 56t
You want to know when the express train's distance is greater than the distance of the local train.
d_express > d_local
56t > 35(t + 1.5)
Solve.
56t > 35(t) + 35(1.5)
56t > 35t + 52.5
56t - 35t > 52.5
21t > 52.5
t > 52.5 / 21
t > 2.5
ANSWER: The express train overtakes the local train after 2.5 hours.
CHECK:
......................... distance ................. rate ...... time
local train …...... d = 35(2.5 + 1.5) ...… 35 ....… 2.5 + 1.5
express train .... d = 56(2.5)................ 56 ....… 2.5
......................... distance ....... rate ...... time
local train …...... d = 140 .......… 35 ....… 4
express train .... d = 140............ 56 ....… 2.5
~~~~~~~~~~
2 With a certain tail wind a jet aircraft arrives at its destination, 1,890 miles away, in 3 hours. Flying against the same wind, the plane makes the return trip in 3 3/8 hours. Find the wind speed and the plane's airspeed.
Variables:
d = distance
r = rate
t = time
c = speed of current
Given: 1,890 miles away
Means: d = 1890
Given: trip with tail wind take 3 hours
Means: t_tailwind = 3
Given: Flying against the same wind, the plane makes the return trip in 3 3/8 hours
Means: t_headwind = 3 3/8
Convert 3 3/8 hours to a decimal.
3 3/8 hours =
3 hours + 3/8 hours =
3 hours + 0.375 hours =
3.375 hours
HINT: Write what you know in a table.
..................... distance ....... rate ...... time
head wind ...... 1890 ........... r - w ...… 3.375
tail wind …...... 1890 ........... r + w ..… 3
total ............... 3780 ........... ---- ........ 6.375
Remember that the distance formula is d = r * t. Apply this to the table.
...................... distance ................... rate ...... time
head wind ...... 1890 = 3.375(r - w) .... r - w ..… 3.375
tail wind …...... 1890 = 3(r + w) ......... r + w .… 3
total ............... 3780 ....................... ---- ........ 6.375
You have 2 equations to use.
1890 = 3.375(r - w)
1890 = 3(r + w)
Solve the first one for r.
1890 = 3.375(r - w)
1890 = 3.375(r) + 3.375(-w)
1890 = 3.375r - 3.375w
1890 + 3.375w = 3.375r
(1890 + 3.375w) / 3.375 = 3.375r / 3.375
(1890 / 3.375) + (3.375w / 3.375) = r
560 + w = r
Plug this into the second equation.
1890 = 3(r + w)
1890 = 3(560 + w + w)
1890 = 3(560 + 2w)
1890 = 3(560) + 3(2w)
1890 = 1680 + 6w
1890 - 1680 = 6w
1890 - 1680 = 6w
210 = 6w
210 / 6 = 6w / 6
35 = w
Plug this into 560 + w = r to find r.
560 + w = r
560 + 35 = r
595 = r
ANSWER: The wind speed is 35 mph; the plane's airspeed is 595 mph.
CHECK:
...................... distance ....................... rate ............ time
head wind ...... 1890 = 3.375(595 - 35) .... 595 - 35 ..… 3.375
tail wind …...... 1890 = 3(595 + 35) ......... 595 + 35 .… 3
total ............... 3780 ............................. ---- ............. 6.375
...................... distance .............. rate ..... time
head wind ...... 1890 = 1890 ......... 560 ..… 3.375
tail wind …...... 1890 = 1890 ......... 630 .… 3
total ............... 3780 .................... ---- ..... 6.375
~~~~~~~~~~
3 The excursion boat on the river takes 2½ hours to make the trip to a point 12 miles upstream and to return. If the rate at which the boat travels in still water is 5 times the rate of the river current, what is the rate of the current?
Variables:
d = distance
r = rate
t = time
c = speed of current
Given: make the trip to a point 12 miles upstream and to return
Means: d = 12
Given: round trip takes 2½ hours
Means: upstream time + downstream time = 2.5
Given: the rate at which the boat travels in still water is 5 times the rate of the river current
Means: r = 5c
HINT: Write what you know in a table.
......................... distance ....... rate ............... time
upstream …....... 12 ................ 5c - c = 4c ..… t
downstream …... 12 ................ 5c + c = 6c .… 2.5 - t
total .................. 24 ................ ---- ................. 2.5
Remember that the distance formula is d = r * t. Apply this to the table.
......................... distan
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{ 2 comments }
Sounds like the 21st was fun!!!! Good to see you had a hoot on the jet boats. we are really excited for you with all the exciting things you are doing. Missed you on your birthday. LOL Mum
I never heard of these “sneaker” waves before either and somehow just from viewing the Pacific Ocean along highway 1 from CA to Oregon and north, the Pacific coast looks so much more rugged and wild compared to the New England coast, except for some places in Maine, which I guess is comparable. But I remember hearing that sailing, for instance, was much harder here on the east coast than on the Pacific and how you really had to be on the alert all the time, because a seemingly calm ocean could change in a flash, the waters becoming really rough, difficult to navigate–that’s what one guy who lives in CA and sails, told me anyway.I think I may have taken those Rogue River jet boats, though we went there a very long time ago, somewhere back in the 80’s and maybe they weren’t around then, but it seemed very similiar. I loved that area–scenery, to die for! Yup, Gus, you’re certainly fortunate to be able to live in one of the most beautiful spots in the US of A.
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