The Macon Psychiatric Institute is interested in redesigning its mental health care delivery system in order to maximize the number of people who can benefit from its services.

Unfortunately, having a large number of patients in residence places a burden on the staff and lengthens patient recovery times. Based on patient records and recovery time estimates by the clinic's staff members, the following mathematical relationship has been developed to describe how the number of patients in residence affects the patient recovery time.

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T = 180 - P

where

T = average patient recovery time in years

P = number of patients in residence

In answering the following questions, think of P as the equilibrium number of patients in residence. We admit a new patient every time we discharge a cured patient in order to keep the number of patients in residence fixed at P.

a. Suppose patients are admitted at a rate that maintains 150 patients in residence at the clinic. What is the average recovery time per patient, and how many patients will recover per year?

b. Using the formula for average patient recovery time, develop an expression for N, which is equal to the number of patients that recover per year as a function of P.

c. Determine the optimal number of patients for the clinic to have in residence if it is interested in maximizing the number of patients that recover per year.