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Sagot :
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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