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At an amusement park there's only enough room for 4500 people to be in it at any time. The manager has also worked out that there needs to be 2800 people in the park to make a profit after all the overhead costs and employee pay. If people are entering the park at a rate of 12 people a minute and there are 850 people in the park currently between how many minutes should the door stay open to let guests in?
(a) Translate the scenario above into a compound inequality involving the number of minutes, m, that the door has been open. Take into account both the fact that there must be a minimum of 2800 people and a maximum of 4500 people.
(b) Rewrite the inequality you found in part (a) using the AND connector and then solve the compound inequality.
(c) Write a solution set as a single statement using interval notation.

Respuesta :

fichoh

Using the concept of inequality, the number of minutes which ensures that the park makes profit while also ensuring that the park isn't overcrowded ls expressed below :

  • m ≤ 304

  • m ≥ 163

  • (m ≥ 163) AND (m ≤ 304)

  • 163 ≥ m ≤ 304

Given the Parameters :

  • Total capacity = 4500

  • Number of people required in park = 2800

  • Entry rate, r = 12 people per minute

  • Initial capacity, c = 850

  • Time, m =?

The number of minute to keep the gates opened :

c + rm ≤ 4500

850 + 12m ≤ 4500

12m ≤ 4500 - 850

12m ≤ 3650

m ≤ (3650÷ 12)

m ≤ 304.16

Hence, the gates should be left opened for at most 304 minutes

B.)

In other to allow atleast 2800 in for profit sake

850 + 12m ≥ 2800

12m ≥ 2800 - 850

12m ≥ 1950

m ≥ 162.5

The gates should be opened for 163 minutes.

C.)

In other to make profit by allowing atleast 2800 people in while also ensuring that not more than 4500 people are in the park ; the inequality representing the number of minutes the gates should be open is :

  • (m ≥ 163) AND (m ≤ 304)

D.)

Expressing as a single interval statement :

  • 163 ≥ m ≤ 304

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