Question 252605
I'll do the first two to get you going.


# 1


Since "t=0 refers to 12:00 a.m", this means that {{{t=10}}} refers to 10:00 a.m. (just add 10 hours to 0)



{{{h=3*sin(pi*t/4)+5}}} Start with the given equation.



{{{h=3*sin(pi*10/4)+5}}} Plug in {{{t=10}}}



{{{h=3*sin(pi*5/2)+5}}} Reduce.



{{{h=3*sin(5pi/2)+5}}} Rearrange the terms



{{{h=3*sin(5pi/2-2pi)+5}}} Subtract {{{2pi}}} from the argument (this is valid because you'll end up on a coterminal angle).



{{{h=3*sin(pi/2)+5}}} Combine like terms.



{{{h=3*1+5}}} Use the unit circle to evaluate the sine of {{{pi/2}}} to get 1.



{{{h=3+5}}} Multiply



{{{h=8}}} Add



So the height of the ocean at 10:00 a.m. is 8 feet. 



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# 2


Remember that {{{cot(x)=1/tan(x)}}} and {{{tan(x)=sin(x)/cos(x)}}}. So this means that {{{cot(3x)=1/tan(3x)=1/(sin(3x)/cos(3x))=cos(3x)/sin(3x)}}} or in short, {{{cot(3x)=cos(3x)/sin(3x)}}}



To find the vertical asymptotes of {{{cot(3x)=cos(3x)/sin(3x)}}}, we'll set the denominator equal to zero and solve for 'x' (since division by zero is undefined).



{{{sin(3x)=0}}} Set the denominator equal to zero.



{{{3x=arcsin(0)}}} Take the arcsine of both sides.



{{{x=0+2pi*n}}} or {{{x=pi+2pi*n}}} Evaluate the arcsine of 0 to get {{{x=0}}} or {{{x=pi}}}. Don't forget to add on multiples of {{{2pi}}} to each solution.



{{{x=(0+2pi*n)/3}}} or {{{x=(pi+2pi*n)/3}}} Divide both sides by 3 to isolate 'x' in each case. 


As a shortcut, you can condense the solution to {{{x=(1/3)pi*n}}} where 'n' is an integer. 



So if {{{x=(1/3)pi*n}}}, where 'n' is an integer, then {{{sin(3x)=0}}}. 



But since {{{0<=x<=pi}}}, this means that we're only going to look at the solutions {{{x=0}}} (for n=0), {{{x=pi/3}}} (where n=1), {{{x=2pi/3}}} (when n=2), and {{{x=pi}}} (when x=3). Note: any other solution is outside the interval {{{0<=x<=pi}}}.



So the four vertical asymptotes of {{{cot(3x)}}} are {{{x=0}}}, {{{x=pi/3}}}, {{{x=2pi/3}}}, and {{{x=pi}}}