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..

Thank you for your help and good explanation too!

Answer 1

The first answer gets close, but a mistake has been made... you find that out when you check the answer. Here is my answer:

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For both parts of the way
[1] ... v1 * t1 = 72
[2] ... v2 * t2 = 72

where v stands for velocity and t for time. Let's start from [2]. Because the total time is 2 hours, t2 = 2 - t1; and we also know that v2 = v1 - 30, so
[3] ... (v1 - 30) * (2 - t1) = 72

This expands to
[4] ... -v1 t1 + 2 v1 + 30 t1 - 60 = 72

Using formula [1], we can reduce this to
[5] ... 2 v1 + 30 t1 = 204
[6] ... v1 = 102 - 15 t1

Substitute this into formula [1],
[7] ... (102 - 15 t1) t1 = 72
[8] ... 15 t1^2 - 102 t1 + 72 = 0

This quadratic equation can be simplified,
[9] ... t1^2 - 2 * 3.4 t1 + 4.8 = 0
[10] ... t1 = 3.4 +/- sqrt (3.4^2 - 4.8)
[11] ... t1 = 3.4 +/- 2.6
[12] ... t1 = 6 hours or 0.8 hours

The first solution does not work, so we conclude t1 = 0.8 hours, t2 = 1.2 hours. The speeds are

v1 = 72/0.8 = 90 mph
v2 = 72/1.2 = 60 mph

Answer 2

Let x be the speed going to the cottage
Then (x - 30) is the speed returning.

Time = DIstance / Rate

So,

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

This can be written as
72(x-30) + 72x = 2(x)(x-30)

This is done by multiplying both sides by the common denominator of (x)(x-30)

Now expand the equation

72x - 2160 + 72x = 2x^2 - 60x

Now bring all terms to one side and set the other side to zero so that you can solve for x.

0 = 2x^2 - 204x - 2160

Using the quadratic formula, you get the values of
x = 111.7 mph
x = -9.7 mph

Obviously, only the first anwer could be true.

So, the car drives approx 111.7mph to the cottage, and 81.7 mph on the way home.

Answer 3

It's pretty straightforward. If the cottage is 72 miles away and her speed driving there x mph, then it will take her
72/x hours to make the trip.

Likewise, when she comes back, she has to drive 30 mph slower so it will that her
72/(x-30) hours.

We're told that the total trip is 2 hours, so....
72/x + 72/(x-30) = 2

And you should be able to take it from here ☺


Doug

Answer 4

You need to set up several equations.
first lets say v1 is the speed going to the cottage and v2 is the speed going to the city. Then we know
v1=v2+30
Also the time it takes to go from city to the cottage is
72/v1=t1
and the time from cottage to city is
72/v2=t2
The total time it takes is t1+t2=2hr.
72/v1+72/v2=2
72/(v2+30)+72/v2=2

you should be able to go from there.

Answer 5

Opps mine is incorrect,the set up that 72/x + 72/(x-30) = 2 is the correct way to go. Sorry.