+1206 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.
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.
+128406 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
