2. A museum has an admission fee of $14 and averages 300 visitors per day. The museum board

decides to raise the fee. Research indicates that for every $1 increase, there would be 10 fewer

visitors per day. Keeping this in mind, the museum committee has found an equation that

represents revenue compared to the number of $1 increases. This relationship can be modeled by

the function: R = - 10x2 + 160x + 4200, where R represents total revenue and x represents the

number of $1 increases above the starting fee of $14.

a) What is the maximum revenue possible for the museum?

b) How many $1 ticket increases will give the museum this maximum revenue?

c) What is the museum's revenue if they do not increase their fee?

The answer to the questions asked is:

a) The maximum possible income for the museum is $4,840 per day.

b) The increase you must make to obtain the maximum income is $8 per ticket.

c) The income of the museum without increasing the fee is $4200.

Step-by-step explanation:

To solve all the questions asked, we must take into account the provided equation:

R = - 10x2 + 160x + 4200

The first question asks for the maximum income, that is, the maximum amount of money that can be obtained by making increases in the rate, despite the loss of people, so you must replace the X in the equation with values until instead of increase decrease:

R = - 10 (1) 2 + 160 (1) + 4200 = 4350 R = - 10 (2) 2 + 160 (2) + 4200 = 4480 R = - 10 (3) 2 + 160 (3) + 4200 = 4590 R = - 10 (4) 2 + 160 (4) + 4200 = 4680 R = - 10 (5) 2 + 160 (5) + 4200 = 4750 R = - 10 (6) 2 + 160 (6) + 4200 = 4800 R = - 10 (7) 2 + 160 (7) + 4200 = 4830 R = - 10 (8) 2 + 160 (8) + 4200 = 4840 R = - 10 (9) 2 + 160 (9) + 4200 = 4830

As you can see, when the increase is $9, the income falls, so the maximum income is $4,840, while it can be seen that it occurs with an increase of $8, which answers the second question.

For the third question, that is, the income of the museum without increasing the fee, you must remember that the variable X is the increase in the fee, therefore, if you want to obtain the income without an increase, it must be given a value of zero to the variable inside the equation, with which we obtain:

R = - 10 (0) 2 + 160 (0) + 4200 = 4200