Question

An electronics bender sells notebook computers the company stocks 100 computers at the beginning of each month. Eighty computers are sold when priced at 1,000 however when priced at 750 all 100 computers are sold assume the relationship between the number of computers c and the price per computer p is linear. Write a linear model that relates the number of computers c to the price p.Use the model to estimate how many computers the company would sell if priced at 1,375 each

Answer 1

$c=-0.08x+160\rightarrow\text{ model}$

b)50 computers

Explanation

Step 1

let c represents the number of computers

let p represents the price per computer

so

$c=f(p)$

so

a)Eighty computers are sold when priced at 1,000

$\begin{gathered} c=80 \\ p=1000 \\ so\text{ } \\ (1000,80) \end{gathered}$

b)when priced at 750 all 100 computers are sold

$\begin{gathered} c=100 \\ p=750 \\ (750,100) \end{gathered}$

now, we have 2 points ( coordinates )

$\begin{gathered} P1(1000,80) \\ P2(750,100) \end{gathered}$

Step 2

find the equation of the line:

a) find the slope

the slope of a line is given by:

$\begin{gathered} \text{slope}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1} \\ \text{where } \\ P1(x_1,y_1) \\ \text{and } \\ P2(x_2,y_2) \end{gathered}$

so, let

P1(1000,80)

P2(750,100)

now, replace

$\begin{gathered} \text{slope}=\frac{y_2-y_1}{x_2-x_1} \\ \text{slope}=\frac{100-80}{750-1000}=\frac{20}{-250}=-0.08 \end{gathered}$

b) equation:

to find the equation we can use the slope-point formula

$\begin{gathered} y-y_1=m(x-x_1) \\ \text{where m is the slope and } \\ (x_1,y_1) \\ is\text{ a point of the line} \end{gathered}$

let

$\begin{gathered} \text{slope}=-0.08 \\ P(1000,80) \end{gathered}$

replace

$\begin{gathered} y-y_1=m(x-x_1) \\ y-80=-0.08(x-1000) \\ y-80=-0.08x+80 \\ \text{add 80 in both sides} \\ y-80+80=-0.08x+80+80 \\ y=-0.08x+160 \end{gathered}$

rewrite usign the defined variables

$c=-0.08x+160\rightarrow\text{ model}$

Step 3

how many computers the company would sell if priced at 1,375 each ?

let

p=1375

replace and calculate

$\begin{gathered} c=-0.08x+160\rightarrow\text{ model} \\ c=-0.08(1375)+160 \\ c=-110+160 \\ c=50 \end{gathered}$

so,

if the price were 1375 each , 50 computers would be sold

I hope this helps you