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One day, I decide to run to the park. On the way there, I run at a rate of \(x^2\) miles per hour for \(3\) hours. On the way back, I take the same path and jog at a slower rate of \(16 - 4x\) miles per hour so that it takes me \(4\) hours to get home. Given that \(x > 0\), what is \(x\)? Express your answer as a common fraction.

 Sep 14, 2018
 #1
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+2

3x^2 = 4(16 - 4x), solve for x
Solve for x:
3 x^2 = 4 (16 - 4 x)

Expand out terms of the right hand side:
3 x^2 = 64 - 16 x

Subtract 64 - 16 x from both sides:
3 x^2 + 16 x - 64 = 0

The left hand side factors into a product with two terms:
(x + 8) (3 x - 8) = 0

Split into two equations:
x + 8 = 0 or 3 x - 8 = 0

Subtract 8 from both sides:
x = -8 or 3 x - 8 = 0

Add 8 to both sides:
x = -8 or 3 x = 8

Divide both sides by 3:
  x = 8/3 =2 2/3 miles per hour.

 Sep 14, 2018
 #2
avatar+98128 
+1

Rate * Time  = Distance

Since the distances are equal, we have that

 

x^2 * 3  = (16 - 4x) * 4

 

3x^2 = 64  - 16x      rearrange as

 

3x^2  + 16x   - 64   = 0    factor

 

(3x - 8) (x + 8)   = 0

 

Set both factors  = 0   and solve for x  and we have

 

x  = 8/3    or  x  = -8

 

Since x > 0, then   x  = 8/3  is correct

 

 

cool cool cool

 Sep 14, 2018

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