I'm going to decode the gibberish in the question and guess the equations are
Rachel:
[tex]h=-16t^2+36t+160[/tex]
Amber:
[tex]h=-16t^2+50t+160[/tex]
These equations tell us a few things not explicitly mentioned in the problem. Â Namely h is in feet, t is in seconds (because of the 16) and the building is 160 feet tall. Â h=0 corresponds to ground level. Â I made the constants of the equations the same because they're both on the twelfth floor. Â The difference between 36t and 50t reflect different vertical components of the velocity of the throw. The positive signs mean they each threw their book upwards. Â Air resistance is ignored.
So we need to solve two quadratic equations and compare the t values.
[tex]0=-16t^2+36t+160[/tex]
[tex]t = \dfrac{-36 \pm \sqrt{36^2 - 4(-16)(160)}}{2(-16)}[/tex]
[tex]t = \dfrac{-36 \pm \sqrt{16(721)}}{-32}[/tex]
[tex]t = \dfrac{9 \pm\sqrt{721}}{8}[/tex]
Only the plus sign gives a positive value of t, so that's our answer for Rachel:
[tex]t_R = \dfrac{9+ \sqrt{721}}{8} \approx 4.48[/tex]
For Amber we get
[tex]0=-16t^2+50t+160[/tex]
[tex]t = \dfrac{-50 \pm \sqrt{50^2 - 4(-16)(160)}}{2(-16)}[/tex]
[tex]t = \dfrac{-50 \pm \sqrt{4(49)(65)}}{-32}[/tex]
[tex]t = \dfrac{25 \pm 7\sqrt{65}}{16}[/tex]
Again we choose the plus sign,
[tex]t_A = \dfrac{25+7\sqrt{65}}{16} \approx 5.09[/tex]
Rachel beat Amber by
[tex]t = t_A - t_R = 5.09-4.58 = 0.51[/tex]
Answer: 0.51 seconds
