Answer to Question #259252 in Algebra for bri

Question

The demand function for a tablet is given by the model p = 200 - 16x² where p is measured in dollars per tablet and x is measured in millions of tablets. If it costs $50 to produce each tablet and a profit of $125 million was derive when 2.5 million tablets were produced. Derive the number of tablets that the company could sell to make the same amount of profit?

Answer 1

demand$p=200-16X^2.....(1)$

p=dollars per tablet

x=millions of tablet

cost for producing each tablet=50

profit=125

revenue=px

$=(200-16x^2)x\\=200x-16x^3$

cost=50x

$profit=RX-CX\\125=200x-16x^3-50x\\125=150x-16^3\\16x^3-150x+125=0\\(2x-5)(8x^2+20x-25)=0$

if

$\\2x-5=0\\x=\frac{5}{2}=2.5$

and if

$8x^2+20x-25=0\\x=2.5$

now the company should sell in the price

$p(2.5)=200-16(2.5)^2\\=200-16\times(\frac{5}{2})^2\\=200-16\times\frac{25}{4}\\=200-4\times 25\\=200-100\\=100$

p(2.5)=100 dollars per tablet.

hence the company should sell 2.5 millions tablet in the price of 100 dollars per tablet to make profit of $125