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**1,2,3. Homework Statement , and work done.**

After Lee gives his little sister Kara a big push on a swing, her horizontal position as a function of time is given by the equation [tex]x(t) = 3cost(t)*e^{-0.05t}[/tex] , where x(t) is her horizontal displacement, in metres, from the lowest point of her swing, as a function of time, t, in seconds.

a) From what horizontal distance from the bottom of Kara's swing did Lee push his sister?

I said 3m.

b) Determine the highest speed Kara will reach and when this occurs.

[tex]x'(t) = -3sin(t)*e^{-0.05t} + 3cos(t)*0.05*e^{-0.05t}[/tex]

[tex]x'(t) = e^{0.05t}(-3sin(t) - 0.15cos(t))[/tex]

[tex]x'(t) = -e^{0.05t}(3sin(t) + 0.15cos(t))[/tex]

So,

[tex]0 = -e^{0.05t}[/tex] or [tex]0 = 3sin(t) + 0.15cos(t)[/tex]

[tex]0 = -e^{0.05t}[/tex]

**DNE**

[tex]0 = 3sin(t) + 0.15cos(t)[/tex]

[tex]-3sin(t) = 0.15cos(t)[/tex]

[tex]-3tan(t) = 0.15[/tex]

[tex]tan(t) = 0.15/-3[/tex]

[tex]t = arctan(0.15/-3)[/tex]

But this gives t = -2.8624; a negative time value.

What did I do wrong?

Part C) How long did it take for Kara's maximum horizontal displacement at the top of her swing arc to diminish to 1m? After how many swings will this occur?

Part D) Sketch the graph <--- I am unsure as to what scale I should use.