Question

A city must determine how to allocate ambulances during the next year. It costs $5000 per year to run an ambulance. Each ambulance must be assigned to one of two districts.

Let xi be the number of ambulances assigned to district i, for i = 1, 2. The average time (in minutes) it takes for an ambulance
to respond to a call from district 1 is 40−3x1; for district 2, the time is 50−4x2.

Gotham City has three goals (listed in order of priority):
Goal 1: At most $90,000 per year should be spent on ambulance service.
Goal 2: Average response time in district 1 should be at most 5 minutes.
Goal 3: Average response time in district 2 should be at most five minutes.

Is it possible to achieve all of these goals? What is the minimum budget required to achieve Goals 2 and 3?

Group of answer choices
A. No. It is not possible to achieve all Goals 1,2,3. The minimum budget required is $120,000.
B. Yes. It is possible to achieve all 3 goals. The minimum budget required is $80000
C. No. It is not possible to achieve all Goals 1,2,3. The minimum budget required is $100,000.
D. No. It is not possible to achieve all Goals 1,2,3. The minimum budget required is $90,000.
E. Yes. It is possible to achieve all 3 goals. The minimum budget required is $90000
F. Yes. It is possible to achieve all 3 goals. The minimum budget required is $85000

Answer 1

After evaluating the constraints, it is not possible to achieve all 3 goals with a budget of $90,000. The minimum number of ambulances needed in each district to meet the response time goals is 12, making the minimum budget required $120,000.

Step-by-step explanation:

To determine if a city can achieve its three stated goals for ambulance allocation, we need to analyze the constraints and perform some calculations:

• Goal 1: Spend no more than $90,000 per year on ambulance service.
• Goal 2: Keep average response time in district 1 under 5 minutes.
• Goal 3: Keep average response time in district 2 under 5 minutes.

Given that it costs $5,000 per year to run an ambulance, we can denote x1 as the number of ambulances in district 1 and x2 as the number in district 2, the total annual cost for running ambulances would be 5000(x1 + x2). This total cost should not exceed $90,000 to satisfy Goal 1.

The response time equation for district 1 is 40−3x1, and for district 2 it is 50−4x2. To satisfy Goals 2 and 3 respectively, we need x1 and x2 to be such that the response times are below or equal to 5 minutes, therefore:

1. 40−3x1 ≤ 5 ⇒ x1 ≥ 35/3 ⇒ x1 ≥ 12 (since we cannot have a fraction of an ambulance)
2. 50−4x2 ≤ 5 ⇒ x2 ≥ 45/4 ⇒ x2 ≥ 12 (since we cannot have a fraction of an ambulance)

Thus, the minimum number of ambulances required for each district to achieve Goals 2 and 3 is 12. Calculating the minimum budget required:

Minimum budget = 5000(x1 + x2) = 5000(12 + 12) = $120,000

Therefore, it is not possible to achieve all three goals with a $90,000 budget.

The minimum budget required to meet response time goals is $120,000, making the correct answer choice A.