import numpy as np

# Define the income function I$($x$)$

def income$($x$)$:

return $(40+$ x$)*(420-10*$ x$)$

# Coefficients of the quadratic equation I$($x$)=$ ax$^2+$ bx $+$ c

# Expanding the income function:

# I$($x$)=40*420-10*40*$ x $+420*$ x $-10*$ x$^2$

# I$($x$)=-10$x$^2+380$x $+16800$

a $=-10$

b $=380$

# The vertex of a parabola given by ax$^2+$ bx $+$ c is at x $=-$b$/(2$a$)$

x$\_$max $=-$b $/(2*$ a$)$

# Calculate the maximum income

max$\_$income $=$ income$($x$\_$max$)$

# Calculate the optimal rental rate

optimal$\_$rate $=40+$ x$\_$max

# Print results

print$($f$"$The rental agency will earn a maximum income of $$\{$max$\_$income:$\mathrm{.}2$f$\}$ when it charges $$\{$optimal$\_$rate:$\mathrm{.}2$f$\}$ per day."$)$