Question

Box office revenue at a multiplex cinema in paris is r(p) = 3600p − 6p3 euros per showing when the ticket price is p euros.

Answer 1

Part A:

Given that Box office revenue at a multiplex cinema in paris is $r(p) = 3600p - 6p^3$ euros per showing when the ticket price is p euros.

When p = 9,

$r(9) = 3600(9) - 6(9)^3 \\ \\ =32,400-6(729)=32,400-4,374 \\ \\ =\bold{28,026 \ euros}$


Part B:

The linear approximation of the change in a function Δf(x) using a value, a, close to x is given by:

$L(a+Δx)=(Δx)f'(a)$

Given that $r(p) = 3600p - 6p^3$, then $r'(p) = 3600 - 12p^2$

Using a = 9, we have:

$r'(9)=3600-12(9)^2 \\ \\ =3600-12(81)=3600-972 \\ \\ =2,628$

Thus, If p is raised by 0.5 euros, then

$\Delta R=(\Delta p)f'(9) \\ \\ =0.5(2,628)=\bold{1,314 \ euros}$



Part C:

The linear approximation of the change in a function Δf(x) using a value, a, close to x is given by:

$L(a+Δx)=(Δx)f'(a)$

Given that $r(p) = 3600p - 6p^3$, then $r'(p) = 3600 - 12p^2$

Using a = 9, we have:

$r'(9)=3600-12(9)^2 \\ \\ =3600-12(81)=3600-972 \\ \\ =2,628$

Thus, If p is lowered by 0.5 euros, then

$\Delta R=(\Delta p)f'(9) \\ \\ =-0.5(2,628)=\bold{-1,314 \ euros}$