Question

The projected sales volume of a video game cartridge is given by the function s(p) = 3000 / (2p + a), where s is the number of cartridges sold, in thousands; p is the price per cartridge, in dollars; and a is a constant. If according to the projections, 100000 cartridges are sold at 10 dollars per cartridge, how many cartridges will be sold at 20 dollars per cartridge?

Answer 1

Answer: 149 cartridges will be sold at 20 dollars per cartridge.

Step-by-step explanation:

Given: The projected sales volume of a video game cartridge is given by the function $s(p)=\dfrac{3000}{2p+a}$, where s is the number of cartridges sold, in thousands; p is the price per cartridge, in dollars; and a is a constant.

Put s(p)=100000, p= 10, we get

$100000=\dfrac{3000}{2(10)+a}\\\\\Rightarrow\ 100=\dfrac{3}{20+a}\\\\\Rightarrow\ 100(20+a)=3\\\\\Rightarrow\ 2000+100a=3\\\\\Rightarrow\ 100a=-1997\\\\\Rightarrow\ a=-19.97$

i.e. $s(p)=\dfrac{3000}{2p-19.97}$

Put p=20, we get

$s(20)=\dfrac{3000}{2(20)-19.97}\\\\=\dfrac{3000}{40-19.97}\\\\=\dfrac{3000}{20.03}\approx149$

Hence, 149 cartridges will be sold at 20 dollars per cartridge.