Rational Equations?

Question

The Chin's Family cottage is 72mi away from their city home. Due to road construction Mrs Chin can drive 30mph faster going to the cottage than she can returning to the city. Given that the trip takes 2 hr, what is Mrs Chin's speed when driving from the cottage to the city?

I have been working on this forever and I just can't figure it out?

that's impossible if she was going 6mph then it would take 12 hours to make one trip

Answer 1

We want to know the speed from the cottage to the city. Let's call that x miles per hour.
Then the speed to the cottage is x+30 miles per hour.

We know the round trip takes 2 hours.

Let y = the driving time from the cottage to the city (in hours).
Then 2 - y = the driving time from the city to the cottage.

Speed times time equals distance. (If I go 10mph and travel 2 hrs, I cover 20 miles of distance).
Hence, we know x*y = 72
and (x+30)*(2-y) = 72

Solving the first equation for x, we get x = 72/y.
Plugging in x from the first equation into the second, we get:

((72/y) + 30) * (2-y) = 72
If you grind through some algebra, this turns into the quadratic equation
5y^2 +14y -24 = 0
And if you solve that (using the quadratic formula), you end up with y = 1.2. or y = -4. Since the driving time can't be negative, ignore the -4 solution and use the 1.2.

So if Mrs Chin drives 72 miles from the cottage to the city in 1.2 hours, she's driving at 72/1.2 = 60 miles per hour.

Double checking the answer: If the return trip takes 1.2 hours, the trip out must take 0.8 hours. 72/0.8 = 90mph,which is 30mph faster than the return trip, so the answer checks out.

Correct answer: Mrs Chin's speed when driving from the cottage to the city is 60 miles per hour.

Hopefully that helps!

Answer 2

It's often helpful for people to draw a picture when solving word problems. We know that there is a cottage and a house. The distance between both places = 72 miles.

What do we know about speeds? We are going x mph if we drive from the cottage to the house. We are going x + 30 (30 mph faster) if we go from the house to the cottage.

We are told that it takes 2 hours to go from the house to the cottage. Writing this as an equation:
72 miles / (x + 30)mph = 2 hours

So now you just basically solve for the x to get the mph for going from the cottage to the house. In this case, its 6mph.

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Update: You're exactly correct -- It would take 12 hours to make the trip through the construction. If you are trying to say that it only takes 2 hours to make the trip through the construction then the solution is trivial -- 72 miles / 2 hours = 36mph.

Update# 2: I think i may have misread your question. I was under the impression that a trip consisted of travel from one destination to the other (not from one to the other and back.)

Assuming that a complete trip involves driving from the cottage, to the house and back to the cottage:

72 miles / ( x + 30 ) mph + 72 miles / x mph = 2 hours

Add these two fractions by making the denominators the same:
(72x + 72( x + 30 )) / ( x(x+30) ) = 2

simplify:

72x + 72( x + 30 ) = 2x( x + 30 )

expand this out to get the x's and constants together. You will arrive at the quadratic:
2x^2-84x-2160=0

Using the quadratic formula...a=2, b = -84, c = -2160

You will find that x = 60

Therefore you can travel 60mph from the cottage to the house.

Answer 3

Right you are, there are more unknowns than equations here!

The equations I used are as follows:

It takes two hours to complete the trip, so it takes t1 to make the first half and t2 for the return, or

t1+t2 = 2 hours

Now, Ms Chin speed coming back is X and driving towards the cottage is (X+30)

The final equations are:

(X+30)*t1 + Xt2 = 144 miles (round trip)
t1+t2 = 2

Two equations, three unknowns

Ask your teacher for one additional piece of information to solve this one!

Good luck.

Answer 4

from what i understand, one way is 72 mi. and the total trip takes 2 hrs. let x=hrs. to city, 2-x=hrs. to cottage; y=speed to city and y+30=speed to cottage. (72/x)=y (eq. 1) and [72/(2-x)=(y+30) (eq.2). by substitution, you get a quadratic equation: 30x^2+84x-144=0. solve for x and you get x=1.2 hrs (going to the city), 2-x=0.8 hrs (going to cottage), y=60 mph (going to city), y+30=90 mph (going to cottage).

Answer 5

I will say 36 mph. from the cottage to the city .