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The annual attendance at the amusement park is initially 2 million people and is increasing at 3% per year. The park’s annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.2 million people per year.
a. Write the equation that represents the food supply. Write the equation represents the park attendance.
The equation that represents food supply is
b. Based on these assumptions, in approximately what year will the amusement park first experience shortages of food?

c. If the park doubled its initial food supply and maintained the rate of increase of 0.2 million people per year, would shortages still occur? In approximately which year?

d. If the park doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur? After how many years would the food supply run out?

Please show work. I will mark brainliest.

Sagot :

semsee45

Answer:

[tex]\textsf{a)} \quad f(t) = 4+0.2t \:\: \textsf{ and }\:\:A(t)=2(1.03)^t[/tex]

b)  77 years

c)  86 years

d)  110 years

Step-by-step explanation:

Part (a)

Given:

  • Initial food supply adequacy = 4 million people
  • Constant annual growth rate = 0.2 million people

As the food supply grows at a constant rate of being adequate for an additional 0.2 million people per year, it can be expressed as a linear function:

[tex]f(t) = 4+0.2t[/tex]

where:

  • f(t) = annual food supply (in millions of people).
  • t = time (in years).

Given:

  • Initial attendance = 2 million people
  • Annual growth rate = 3%

As the annual attendance increases by 3% per year, it can be expressed as an exponential function:

[tex]A(t)=2(1.03)^t[/tex]

where:

  • A(t) = annual attendance (in millions of people).
  • t = time (in years).

Part (b)

Graph the two functions (see attached) and locate the value of t for which A(t) > f(t) for the first time.

From inspection of the graph, the two functions intersect at t ≈ 77.  So after approximately 77 years the food supply will not be enough for the number of people attending the amusement park.  Therefore, after approximately 77 years, the park will first experience shortages of food.

Part (c)

If the park doubles its initial food supply and maintains the rate of increase of 0.2 million people per year, the new equation would be:

[tex]f(t) = 8+0.2t[/tex]

Again, graph the new function against A(t) and find the point where the two functions intersect.  A(t) = f(t) at approximately t = 86 so the park will first experience food shortages after 86 years.  So doubling the initial food supply delays the eventual food shortage by only c. 9 years.

Part (d)

If the park doubled the rate at which its food supply increases in addition to doubling its initial food supply, the new equation would be:

[tex]f(t) = 8+0.4t[/tex]

Again, graph the new function against A(t) and find the point where the two functions intersect.  A(t) = f(t) at approximately t = 110 so the park will first experience food shortages after 110 years.  So doubling the initial food supply and doubling the rate delays the eventual food shortage by only c. 33 years compared to the initial parameters.  Shortages would still occur, but it would be later in approximately 110 years' time.

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