Advanced algebra word problems? I have a midterm next week and have been struggling with these????

Question

If someone could take me through the steps of how to set up the equation and solve these word problems, would really help me so I can get this concept down for my midterm next week? Here goes:

1)Rachel allows herself one hour to reach a sales appointment 50 miles away. After she has driven 30 miles she realizes that she must increase her speed by 15 mph in order to get there on time. What was her speed for the first 30 miles?

2) The main engine of a rocket burns for 60 seconds on the fuel in the rocket's tank, while the auxillary engine burns for 90 seconds on the same amount of fuel. How long do both engines burn if they are operated together on the same tank of fuel?

Answer 1

Question 1
Distance 1
30 miles ; speed = x mph ; time = t1 h
t1 = 30 / x
Distance 2
20 miles ; speed = (x + 15) mph ; time = t2 h
t2 = 20/(x + 15)
t = t1 + t2 where t = 1
1 = 30 / x + 20 / (x + 15)
x.(x + 15) = 30.(x + 15) + 20.x
x² + 15.x = 30.x + 450 + 20.x
x² - 35.x - 450 = 0
x = [35 ± √ (35² + 1800) ] / 2
x = [ 35 ± 55] / 2 (using +ve value only)
x = 90 / 2
x = 45 mph
Speed for first 30 miles = 45 mph

Question 2
Let amount of fuel = x litres
main engine rate = x / 60 litres / sec
Auxiliary engine rate = x / 90 litres / sec
Fuel per sec = x / 60 + x / 90
Fuel per sec = 3x /180 + 2x /180 = 5x/180
Time = x / (5x /180) sec = 180 / 5 sec = 36 sec

Answer 2

Solution to no. 1

the formula for velocity is v = d / t where d is distance travelled and t is the time taken.

from the problem, it mentioned that she usually travel the 50 miles in just 1 hour so that she will not be late, so she must have a total velocity of,

total velocity = 50 / 1 or 50mph

Therefore for 30miles her speed is:

= 30 / t

She increased her speed after 30miles by 15mph, so

(30 / t) + 15 = 50mph

30 / t = 50 - 15

30 / t = 35

30 = 35 t

t = 30 / 35

t = 6 / 7 hour, this is the time she took after travelling for 30 miles, therefore her velocity is,

= 30 / t

= 30 / (6/7)

= 30 (7 / 6)

= 210 / 6

= 35mph

Answer 3

since this is math it is thoeretical so lets assume rachel's appt is on the side of the road at mile marker 50 and she parks her car a marker 0 and she has magic brakes and accelerator and there are no stoplights along the way. she wants to drive 50mph for 1 hour to reach her appt, so she takes off but then 30 miles down the road she realizes she has to pee and fix her hair and that will take at least 5 minutes. Rachel is great at time management. she knows she'll have to speed up to 65 to make up the time she'll need for the lavatory.

2) f=60m so m=f/60
f=90a so a=f/90
now we want to know time for
(a+m)(sec)= f/90 + f/60 = 5f/180
180 sec = 5f
f=180/5=36 seconds

Answer 4

X=distance
V=speed
T=time

X=V*T
30=V*T and
20=(V+15)*(1-T)

so 20-V-15= -T*(V+15)
-(5-V)/(V+15) = T
T=(V-5)/(V+15)

so 30=V*(V-5)/(V+15)
30V+450=V^2-5V
V^2-35V-450=0

V=45

2) 1/60 + 1/90 = 5/180 = 1/36, so 36 seconds

Answer 5

the first question lacks some information...should have gievn the minimum time to reach the 50miles destination...

about question two...i dont look at it as it might also lack some quantity