MathematicsOpen Ended
In a particular country the annual availability of head lettuce (such as iceberg) has been decreasing in recent years, while the availability of leaf lettuce (such as romaine) has been increasing. For the years 2000 through 2019, the function
y = - 0.56x + 20.02 approximates the annual country's per capita availability of
head lettuce in pounds, and the function y = 0.3x + 8.2 approximates the
annual country's per capita availability of leaf lettuce in pounds. For both functions, × is the number of years after 2000. Complete parts a and b
a. Explain how the given function for head lettuce verifies that the availability of head lettuce has decreased while the given function for leaf lettuce verifies that the availability of leaf lettuce has increased b. Based on this information, determine the year in which the per capita availability of head lettuce equaled that of leaf lettuce
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Explanation:
a. The given function for head lettuce, y = -0.56x + 20.02, has a negative slope (-0.56), which means that as x (the number of years after 2000) increases, y (the per capita availability of head lettuce) decreases. This verifies that the availability of head lettuce has been decreasing in recent years. On the other hand, the given function for leaf lettuce, y = 0.3x + 8.2, has a positive slope (0.3), which means that as x increases, y (the per capita availability of leaf lettuce) increases. This verifies that the availability of leaf lettuce has been increasing in recent years.
b. To determine the year in which the per capita availability of head lettuce equaled that of leaf lettuce, we need to find the point where the two functions intersect. Setting the two functions equal to each other, we get:
-0.56x + 20.02 = 0.3x + 8.2
Solving for x, we get:
x = 18.5
This means that the per capita availability of head lettuce equaled that of leaf lettuce 18.5 years after 2000. To find the year, we add 18.5 to 2000, which gives us the year 2018. Therefore, the per capita availability of head lettuce equaled that of leaf lettuce in the year 2018.
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