Question

An aircraft manufacturer wants to determine the best selling price for a new airplane. The company estimates that the initial cost of designing the airplane and setting up the factories in which to build it will be 500 million dollars. The additional cost of manufacturing each plane can be modeled by the function below, where x is the number of aircraft produced and m is the manufacturing cost, in millions of dollars.
m(x) = 20x - 5x^3/4 + 0.01x^2
The company estimates that if it charges a price p (in millions of dollars) for each plane, it will be able to sell x(p) = 320 - 7.7p planes.
(a) Find the cost, demand (or price), and revenue functions.
C(x) =
p(x) =
R(x) =
(b) Find the production level that maximizes profit.
Planes =
(c) Find the associated selling price of the aircraft that maximizes profit.
= $_______ million
(d) Find the maximum profit.
= $ _______ million

Answer 1

Answer:

$(a)$ $C(x)=500+20x-5x^{\frac{3}{4}}+0.01x^2$

    $p(x)=320-7.7p$

    $R(x)=(320-7.7p)p=320p-7.7p^2$

$(b)$ $x=82 \text{planes}$

$(c)$ $p=\ 30.91 M\;\; \text{per plane}$

$(d)$ maximum profit $=\ 15.90M$

Step-by-step explanation:

Given that,

The company estimates that the initial cost of designing the aeroplane and setting up the factories in which to build it will be 500 million dollars.

The additional cost of manufacturing each plane can be modelled by the function.

$m(x)=20x-5x^{\frac{3}{4}}+0.01x^2$

$(a)$  Find the cost, demand (or price), and revenue functions.

   $C(x)=500+20x-5x^{\frac{3}{4}}+0.01x^2$

   $p(x)=320-7.7p$

   $R(x)=(320-7.7p)p=320p-7.7p^2$

$(b)$  Find the production level that maximizes profit.

    $f=R(x)-C(x)$

 $\Rightarrow f=320p-7.7p^2-(500+20x-5x^{\frac{3}{4}}+0.01x^2)$

$\Rightarrow df=320dp-15.4pdp-20dx+5(\frac{3}{4} )x^{\frac{-1}{4} }dx-0.02xdx$

     $x=320-7.7p$

     $p=\frac{320-x}{7.7}$

    $\frac{dp}{dx} = \frac{-1}{7.7}$

$\frac{df}{dx}=\frac{320}{-7.7} -\frac{15.4(320-x) }{7.7(\frac{-1}{7.7} )}-20+5\frac{3}{4} x^{\frac{-1}{4}} -0.02x=0$

    $\Rightarrow -41.5584+83.1169-0.2597x-20+3.75x^{\frac{-1}{4} }-0.02x=0$

   $\Rightarrow 21.5585+3.75x^{\frac{-1}{4} }-0.279x=0$

   $\Rightarrow x=82 \text{planes}$

$(c)$  Find the associated selling price of the aircraft that maximizes profit.

  $p=\frac{320-82}{7.7}$

$\Rightarrow p=\ 30.91 M\;\; \text{per plane}$

$(d)$  Find the maximum profit.

Manufacturing cost of one plane is:

$m(1)=20-5+0.01$

         $=\ 15.01 M$

maximum profit $=\ (30.91-15.01)M$

                           $=\ 15.90M$