Finding a formula for the number of patrons as a function of profit.

[kingpie]

The only question I need help with is part (C) of this question.

A theater manager graphed weekly profits as a function of the number of patrons and found that the relationship was linear. One week the profit was $10,246 when 1376 patrons attended. Another week 1585 patrons produced a profit of $12,022.50

A.) Find a formula for weekly profit, y, as a function of the number of patrons, x.
y=$-1450+8.5x
B.) What is the break-even point (the number of patrons for which there is zero profit)? Round your answer to the nearest integer.

170.6
C.) Find a formula for the number of patrons as a function of profit. Exact answer.

D.) If the weekly profit was $15,575.50, how many patrons attended the theater?
2003

 

[Deleted member 4993]

The only question I need help with is part (C) of this question.

A theater manager graphed weekly profits as a function of the number of patrons and found that the relationship was linear. One week the profit was $10,246 when 1376 patrons attended. Another week 1585 patrons produced a profit of $12,022.50

A.) Find a formula for weekly profit, y, as a function of the number of patrons, x.
y=$-1450+8.5x
B.) What is the break-even point (the number of patrons for which there is zero profit)? Round your answer to the nearest integer.

170.6
C.) Find a formula for the number of patrons as a function of profit. Exact answer.

D.) If the weekly profit was $15,575.50, how many patrons attended the theater?
2003

Have you worked with "inverse" function?

You need to find "x" as function of "y".

 

[lev888]

Double check your number of patrons answer. Can't have fractional people running around.

 

[Aliabla]

Let's make the profit be y and number of patrons be x since it has a linear relation, we define a general linear equation as
y= ax +b
where a and b are some constant from the question we get that for
y= 10246 , x= 1376
and y= 12,022.50, x= 1585 m= (12,022.50-10246)/(1585-1376)= m=8.5

y=8.5x+b

Now we plug in one of the two combinations we have of patrons and money

10246 = (8.5)(1376) + b
b = -1450

y=8.5x-1450

So now our function is:

Money = (8.5)(Patrons) - 1450

(c) To find the break-even point, we want to set "money" equal to 0 to mean no profit

0 = (8.5(Patrons) - 1450

Patrons = about 171

To find this function, we just solve for "Patrons" in our above linear equation

just solve y=8.5x-1450 to get your formula



lastly x=(15575.50+1450)/(8.5)=2003

 

[JeffM]

Let's make the profit be y and number of patrons be x since it has a linear relation, we define a general linear equation as
y= ax +b
where a and b are some constant from the question we get that for
y= 10246 , x= 1376
and y= 12,022.50, x= 1585 m= (12,022.50-10246)/(1585-1376)= m=8.5

Correct. So you conclude correctly that

y=8.5x+b

Now we plug in one of the two combinations we have of patrons and money

10246 = (8.5)(1376) + b
b = -1450

Correct again. Therefore

y=8.5x-1450

So now our function is:

Money = (8.5)(Patrons) - 1450

(c) To find the break-even point, we want to set "money" equal to 0 to mean no profit

0 = (8.5(Patrons) - 1450

Patrons = about 171

Correct for the third time.

To find this function, we just solve for "Patrons" in our above linear equation

just solve y=8.5x-1450 to get your formula

But you did not write the formula so you might not get credit on a test.

[MATH]y = 8.5x - 1450 \implies y + 1450 = 8.5x \implies x = \dfrac{y + 1450}{8.5}.[/MATH]
However, you might get credit or at least partial credit because you clearly knew what the formula is as shown by

lastly x=(15575.50+1450)/(8.5)=2003

which is correct.

 

[Aliabla]

I was trying to let the OP figure out question c on there own!!