A movie theater owner would like to determine the best price to charge for popcorn. She has determined that during a typical movie, if she charges $4 for a bag, she will sell 200 bags, and if she charges $6 for a bag, she will sell 120 bags.

a)Find a linear model, using price as the independent variable, that models the numbers of bags sold vs price. Be sure to label what your variables represent.

b) Find an exponential model, using price as the independent variable, that models the numbers of bags sold vs price. Be sure to label what your variables represent.

1.For linear model

Given: (4,200) (6,120)

(x₁,y₁) = (4,200)

(x₂,y₂) = (6,120)

let us get the slope:

m = (y₂ - y₁)/ (x_{2 -} x₁)

m = (120 - 200)/ (6 - 4)

m = -40

The equation of the line is:

y - y₁ = m (x - x₁)

y -200 = -40(x - 4)

y -200 = -40x + 160

y = -40x + 160 + 200

y = -40x + 360

Therefore the linear model is

f(x) = -40x + 360

where:

f(x) = no. of bags she will sell

x = price

 

Example,

f(x) = -40x + 360

f(4) = -40(4) + 360

f(4) = 200

 

f(6) = -40(6) + 360

f(6) = 120

 

2.For exponential model

Given: (4,200) (6,120)

We have a formula of y = ab^x

so,

200 = ab⁴ eq.1

120 = ab⁶ eq.2

 

200 = ab⁴

200/b⁴ = a

 

substitute to equation 2

120 = ab⁶

120 = (200/b⁴ )(b⁶ )

120 = 200b²

120/200 = b²

0.6 = b²

b = 0.7745

 

Now that we have the value of b,

200 = ab⁴ eq.1

200 = a(0.7745)⁴

200 = a(0.3598)

200/0.3598 = a

a = 555.83

 

Therefore the exponential model is:

f(x)=(555.83)(0.7745)^x

?where:

f(x) = no. of bags she will sell

x = price

 

Example:

f(4)=(555.83)(0.7745)⁴

f(4)=200

 

f(6)=(555.83)(0.7745)⁶

f(6)=120

 

which correct (4,200) and (6,120)