algebra 2 help?

Question

1. Joe traveled against the wind in a small plane for 3hr. the return trip with the wind took 2.8 hr. find the speed of the wind if the speed of the plane in still air is 180mph
2. If Joe can paint a house in 6 hr, and Sam can paint the same house in 8 hr, how long does it take them to do it together
3. Sweage treatment plant has two inlet pipes to its settling pond. One can fill the pond in 10 hr, the other in 12 hr. if the first pipe is open for 5 hr and then the second pipe is opened, how long will it take to fill the pond?
4. How many gallons of pure alcohol should be mixed with 20 gal of a 15% alcohol solution to obtain a mixture that is 25% alcohol?
5. Travis Hayes needs 10% hydrochloric acid for a chemistry experiment. How much 5% acid should be mixed with 60 mL of 20% acid to get a 10% solution?

Answer 1

1.) Set up the equation to take everything into account:
d = (v + w)t, because the total velocity is plane + the wind speed.
d = (v - w)t, because the total velocity is plane - the wind speed.

Since d is the same for both (it's a round trip, you can rewrite it as:
(v + w)t1 = (v - w)t2
(180 + w) * 2.8 = (180 - w) * 3
504 + 2.8w = 540 - 3w
36 = 5.8w
w = 6.207 mph (solution!)

check:
(180 + w) * 2.8 = (180 - w) * 3
(180 + 6.207) * 2.8 = (180 - 6.207) * 3
186.207 * 2.8 = 173.793 * 3
521.379 = 521.379 (check!)

2.) Set up equations to figure out how much work is done in 1 hour.
Joe paints a house in 6 hours, so he paints 1/6 of a house in 1 hour.
Sam paints a house in 8 hours, so he paints 1/8 of a house in 1 hour.
1/6 house/hr * t + 1/8 house/hr * t = 1 house painted
t * (1/6 + 1/8) = 1
t * (8/48 + 6/48) = 1
t * 14/48 = 1
t = 48/14
t = 24/7 = 3 3/7 hr. (check!)

check:
1/6 * t + 1/8 * t = 1
1/6 * 24/7 + 1/8 * 24/7 = 1
4/7 + 3/7 = 1 (check!)

3.) This is similar to #2:
If pipe 1 can fill the pond in 10 hours, it fills 1/10 of the pond in 1 hour.
After 5 hours, pipe #1 has filled 5 * 1/10th of the pond = 1/2 of the pond.
Pipe 2 can fill the pond in 12 hours, so it fills 1/12 of the pond in 1 hour.

1/10 t + 1/12 (t-5) = 1 (t-5 is used for pipe 2, because it was not open the first 5 hours)
t/10 + t/12 - 5/12 = 1
t/10 * 6/6 + t/12 * 5/5= 17/12
6t/60 + 5t/60 = 17/12
11t/60 = 17/12
t = 17/12 * 60/11
t = 85/11 = 7 8/11 hr (solution)!

4.) Set up the equation:
20 gal * .15 alcohol/ga l+ a gal * 1 alcohol/gal = (20+a) gal * .25 alcohol / gal
20 * .15 + a = (20 + a)* .25
3 + a = 5 + .25a
2 = .75a
a = 8/3 gallons (solution)

Check:
20 gallons * .15 + 2 2/3 gallons * 1 = 22 2/3 gallons * .25
3 + 2 2/3 = 5 2/3
5 2/3 = 5 2/3 (check!)

5.) Similar to #4:
.05 * x + .20 * 60 = .10 (60 + x)
.05x + 12 = 6 + .1x
6 = .05x
x = 120 mL (solution!)

check:
.05 * 120 + .20 * 60 = .10 (60 + 120)
6 + 12 = 18 (Check!)

Answer 2

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