Question

A local aquarium found that if the price of admission was $14, the attendance was about 1650 customers per day. When the price of admission was dropped to $8,attendance increased to about 1750 per day. Write a linear equation for the attendance in terms of the price, p. (A = mp + b)

Answer 1

We want to calculate the linear equation that models the attendance by using the admission price.

So, the model would look like this

`A=mp+b`

where A is the attendance, p is the price, m is the slope of the line and b is the y intercept. We know that when p=14 then A=1650, which can be summarized by writing (14,1650). Also, we know that when p=8, then A=1750. So we write (8,1750).

To write the linear equation, we need to determine m and b using this points. To do so, recall that given points (a,b) and (c,d), the slope of the line is given by the formula

`m=(d-b)/(c-a)=(b-d)/(a-c)`

So, in our case we have a=14, b=1650, c=8 and d=1750. So we have

`m=(1750-1650)/(8-14)=(100)/(-6)=-(50)/(3)`

So, so far we have the following

`A=-(50)/(3)p+b`

Now, recall that whenever p=8, then A=1750. So if we replace this values in our equation we get

`1750=-(50)/(3)\cdot8+b`

If we multiply both sides by 3, we get

`-50\cdot8+3b=1750\cdot3`

now, if we add 50*8 on both sides, we get

`3b=1750\cdot3+50\cdot8`

Finally, we divide both sides by 3, so we get

`b=(1750\cdot3+50\cdot8)/(3)=(5650)/(3)`

So our equation becomes

`A=-(50)/(3)\cdot p+(5650)/(3)`

To check that this answer is correct, we will use the other point. That is, we will replace p=14 and check if it leads to A=1650. NOte that