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

$m=\frac{d-b}{c-a}=\frac{b-d}{a-c}$

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

$m=\frac{1750-1650}{8-14}=\frac{100}{-6}=-\frac{50}{3}$

So, so far we have the following

$A=-\frac{50}{3}p+b$

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

$1750=-\frac{50}{3}\cdot8+b$

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

$b=\frac{1750\cdot3+50\cdot8}{3}=\frac{5650}{3}$

So our equation becomes

$A=-\frac{50}{3}\cdot p+\frac{5650}{3}$

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

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