## project one 
# V= l*w*h
# V=(3-2*x)*(8-2*x)*x
# V=(4*x^2-22*x+24)*x
#V= (4*x^3-22*x^2+24*x)
V(x)= 4*x^3-22*x^2+24*x
diff(V(x),x)

Vquad=solve([4*x^2-44*x+24==0],x)
print(Vquad)

# x equates to .57557 and 10.4244
# which means the x for the equation would be .57557 because if it was 10 then the length would be negative and that cannot happen
#so the maximum volume is .57557
print('x=.57557 and x=10.4244')
print('which means that x=.57557 because the volume cannot be negative cannot be negative')
s=simplify(8*3*0.57557)
print('therefore the maximum volume of the box is', s)

## project two
p=var('p')
x=1000-p
p=1000-x

#C(x)=cost
C(x)= 3000+20*x
#R(x)=revenue
R(x)=x*(1000-x)

#the revenue
print(R(x))
p1=plot(R(x), -100,1000, ymin=-4000, ymax=300000, gridlines='minor', linestyle='-', color='blue')
p1.show()

# the profit
P(x)=R(x)-C(x)
print(P(x))
p2=plot(P(x), -100,100, ymin=-4000, ymax=30000, gridlines='minor', linestyle='-', color='red')
p2.show()

#maximize production with 500 units per day
print('the cost of 500 units a day',C(500))

#maximum profit
pf(x)=diff(P(x),x)
solve(pf(x),x)
print('maximum profit is 490')

##project three
##create p(x)
##.9906y(1000-y)
# = y((.9906*1000)/a)-(.9906y/b))
y=var('y')
p=.9906*y*(1000-y)
p1=plot_slope_field(p,(x,0,100),(y,-10,1000), headlength=5, headaxislength=5, color='blue')
p1.show()

p(x)=990.6/(.9906+(990.6-.9906)*exp(-.9906*x))
p2= plot(p(x), 0, 100, ymin=0, ymax=1000, gridlines= 'minor', color='red')
p2.show()
show(p1+p2)
print('the number of sick kids in six days would be',p(6))
print('since this is an exponential function all students will eventually get infected, unless someone finds a cure')
print('since there is no cure found yet, there is no recovered person in the data yet.However, we can see the numbers plateau at 1000')

## project 4
pw= piecewise([((0, pi/2),-1), ((pi/2, pi), 2)])
print(pw)
p=plot(pw,x,-10,10, ymin=-10, ymax=10, gridlines='minor', linestyle='-', color='red')
p.show()

fsp1=pw.fourier_series_partial_sum(5)
print(fsp1)
p1=plot(fsp1,-3,3, ymin=-3, ymax=3, gridlines='minor', linestyle='-', color='blue')
show(p1)

fsp2=pw.fourier_series_partial_sum(10)
print(fsp2)
p2=plot(fsp2,-3,3, ymin=-3, ymax=3, gridlines='minor', linestyle='-', color='red')
fsp3=pw.fourier_series_partial_sum(15)
print(fsp3)
p3=plot(fsp,-3,3, ymin=-3, ymax=3, gridlines='minor', linestyle='-', color='green')
show(p1+p2+p3)
print('we can notice that it fluctuates in the same general areas and will most likely continue that pattern')

pw2= piecewise([((-1,0),-x),((0, 1),x)])
print(pw2)
fsp4=pw2.fourier_series_partial_sum(10)
p4=plot(fsp4,-3,3, ymin=-3, ymax=3, gridlines='minor', linestyle='-', color='orange')
p4.show()