4. Linda drove to Eureka. She drove 240 miles. On the way back,

she was having car troubles and she had to drive 20 mi/hr slower. If

it took her 2 hours longer to get back, how fast was she driving on

the trip there, and on the trip back?

 

The first step in solving word problems is to define the

unknown. In this problem, there are two unknowns, Linda's speed

driving to Eureka and her speed driving back. When it says that she

was driving 20 mi/hr slower on the way back, that gives us the

speed on the way back as a function of her speed on the way there.

So we

 

let x = her speed there.

then

x - 20 = her speed back

 

Your author suggests that we make use of d = rt tables. We

know that the distance both there and back is 240 miles. We also

have expressions for the rate. That gives us this much of the table.

 

 

Once we have two columns for the table filled in we can use the

appropriate form of the d = rt equation to fill ion the third column.

Since the unfilled column is the one for time, we need to use the

equation

 

 

to fill in the time. That enables us to fill in the table as follows.

 

 

 

Now the sentence "It took her 2 hours longer to get back." can

be translated as

 

 

Since we now have an equation, we can take the steps to solve

it. First, clear denominators. The smallest common denominator is

x(x-20), so multiply both sides by that.

 

 

The x(x-20) needs to be distributed through the two terms on

the right.

 

 

 

This will clear all of our denominators. We assemble the

surviving factors.

 

 

 

Remove parentheses.

 

 

and first notice that we can subtract 240x from both sides of the

equation. That leaves us with the quadratic

 

 

We already have a 0 on one side, so we are ready to see if it

factors. First notice that there is a common factor of 2 in all the terms

which can be factored out.

 

 

That makes it a little easier to factor the expression inside. It will

factor because there are factors of 2400 which have a difference of

20: 40 and 60.

 

 

 

This product to be equal to 0, if and only if one of these factors

is equal to 0. The factor of 2 is never equal to 0, so one of the other

factors must be equal to zero. Set them equal 0

 

 

 

Check:

If she was driving at 60 mi/hr going to Eureka, then it would

have taken her 4 hours to get there. If she drove 20 mi/hr slower on

the way back, then she was driving at only 40 mi/hr, and it would

have taken her 6 hours, which is 2 hours longer than it took her to

get there.

 

But if she was driving at -40 mi/hr, then the computation for

the time it would take would be

 

t = 240/(-40)

 

t = - 6

 

This could be interpreted as meaning that if she was driving at

-40mi/hr, then she was driving 40 mi/hr backward and at that rate

she would have been in Eureka 6 hours ago. 20 mi/hr slower than

-40 mi/hr would be - 60 mi/hr and at that rate she would have been

in Eureka only 4 hours ago which is 2 more hours than -6 hours.

However, many authors will simply reject negative velocities as

being meaningless.

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