SOLUTION: Karen went on a 48 mile trip to a soccer game. On the way back, due to road construction she had to drive 24 miles per hour slower. This made the trip take 1 hour longer. How fast

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Question 1150320: Karen went on a 48 mile trip to a soccer game. On the way back, due to road construction she had to drive 24 miles per hour slower. This made the trip take 1 hour longer. How fast did she drive to the soccer game?
Found 2 solutions by jim_thompson5910, MathTherapy:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x = Karen's speed, in mph, going from her house to the soccer game
y = time it took, in hours, to go from Karen's house to the soccer game
d = 48 is the distance traveled in miles

When Karen goes from her house to the game, then we can set up this equation below:
distance+=+rate%2Atime

d+=+r%2At

48+=+x%2Ay Plug in d = 48, r = x, and t = y.

x%2Ay+=+48

%28x%2Ay%29%2Fx+=+48%2Fx Divide both sides by x to isolate y.

y+=+48%2Fx We'll use this equation later

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When heading back home, she takes 1 hour longer. So instead of driving for y hours, she drives for (y+1) hours. This is because her speed drops from x mph to (x-24) mph. The distance d is still the same at 48 miles.

d+=+r%2At

48+=+%28x-24%29%2A%28y%2B1%29 Plug in r = (x-24) and t = (y+1)

48+=+%28x-24%29%2A%2848%2Fx%2B1%29 Next, plug in y = 48/x, which was that equation we found earlier. At this point, we need to solve for x. But first let's do a bit of algebra to simplify this equation.

%28x-24%29%2A%2848%2Fx%2B1%29+=+48

x%2A%2848%2Fx%2B1%29-24%2848%2Fx%2B1%29+=+48 Use the distributive property

48%2Bx-1152%2Fx-24+=+48 Use the distributive property again

48-24%2Bx-1152%2Fx+=+48

24%2Bx-1152%2Fx+=+48

x%2A%2824%2Bx-1152%2Fx%29+=+x%2A48 Multiply both sides by x so that you clear out the fraction.

24x%2Bx%5E2-1152+=+48x Note how the fraction is eliminated after using the distributive property.

24x%2Bx%5E2-1152-48x+=+48x-48x Subtract 48x from both sides

x%5E2-24x-1152+=+0

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In order to solve x%5E2-24x-1152+=+0, we can use the quadratic formula.

Compare that equation to the general quadratic ax%5E2%2Bbx%2Bc+=+0 and we see that a = 1, b = -24, c = -1152.

Plug those a,b,c values into the quadratic formula below. Then simplify.

x+=+%28-b%2Bsqrt%28b%5E2-4ac%29%29%2F%282a%29 or x+=+%28-b-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-24%29%2Bsqrt%28%28-24%29%5E2-4%281%29%28-1152%29%29%29%2F%282%281%29%29 or x+=+%28-%28-24%29-sqrt%28%28-24%29%5E2-4%281%29%28-1152%29%29%29%2F%282%281%29%29

x+=+%2824%2Bsqrt%285184%29%29%2F%282%29 or x+=+%2824-sqrt%285184%29%29%2F%282%29

x+=+%2824%2B72%29%2F%282%29 or x+=+%2824-72%29%2F%282%29

x+=+%2896%29%2F%282%29 or x+=+%28-48%29%2F%282%29

x+=+48 or x+=+-24

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A negative speed does not make sense, so we ignore x = -24.
The only practical solution is x = 48.

So she drives from her house to the soccer game at 48 mph.
If x = 48, then y = 48/x = 48/48 = 1. This means it takes her 1 hour to drive from her house to the soccer game.

Also, if x = 48, then x-24 = 48-24 = 24 is her speed coming back home. The time it takes to drive back home is t = d/r = 48/24 = 2 hours, which fits with the description that she took one hour longer (compared to her journey from her house to the soccer game).

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Answer: 48 miles per hour

Answer by MathTherapy(10557) About Me  (Show Source):
You can put this solution on YOUR website!
Karen went on a 48 mile trip to a soccer game. On the way back, due to road construction she had to drive 24 miles per hour slower. This made the trip take 1 hour longer. How fast did she drive to the soccer game?
EVERYTHING that the other person did is way too COMPLEX and WAY TOO time-consuming.

If I were you, I'd simply IGNORE it!



This is the correct way to do the problem:

Let the outgoing speed be S

Then the return speed = S - 24

The return trip took 1 hour longer, so we get the following TIME equation: matrix%281%2C3%2C+48%2FS%2C+%22=%22%2C+48%2F%28S+-+24%29+-+1%29 

48(S - 24) = 48S - S(S - 24) ------ Multiplying by LCD, S(S - 24)

matrix%281%2C3%2C+48S+-+48%2824%29%2C+%22=%22%2C++48S+-+S%5E2+%2B+24S%29

matrix%281%2C3%2C+S%5E2+-+24S+-+48S+%2B+48S+-+%221%2C152%22%2C+%22=%22%2C+0%29

matrix%281%2C3%2C+S%5E2+-+24S+-+%221%2C152%22%2C+%22=%22%2C+0%29

(S - 48)(S + 24) = 0

S, or highlight_green%28matrix%281%2C5%2C+OUTGOING%2C+speed%2C+%22=%22%2C+48%2C+mph%29%29       OR       S = - 24 (ignore)

That's IT!! No NOVEL has to be written on this!