Kudzu is a vine that was introduced to the United States from Japan in 1876 as an ornamental plant. Starting in the year 1935 , kudzu was planted throughout the Southeast to combat soil erosion. By 1946, kudzu covered approximately 3 million acres of land in the Southeast. Sixty years later, about 7 million acres were covered by kudzu. Which of the following functions best models the amount of kudzu, in millions of acres, t years after 1946? Choose 1 answer: (A) K(t)=3*(1.014)^(t) (B) K(t)=3*(0.014)^(t) (c) K(t)=3+(1.014)t (D) K(t)=3+(0.014)t

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Kudzu is a vine that was introduced to the United States from Japan in 1876 as an ornamental plant. Starting in the year 1935 , kudzu was planted throughout the Southeast to combat soil erosion. By 1946, kudzu covered approximately 3 million acres of land in the Southeast. Sixty years later, about 7 million acres were covered by kudzu. Which of the following functions best models the amount of kudzu, in millions of acres,  t years after 1946? Choose 1 answer: (A)  K(t)=3*(1.014)^(t) (B)  K(t)=3*(0.014)^(t) (c)  K(t)=3+(1.014)t (D)  K(t)=3+(0.014)t

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Understanding the Growth: To determine which function best models the growth of kudzu over time, we need to understand the nature of the growth. We are given two data points: in 1946, kudzu covered 3 million acres, and 60 years later, it covered 7 million acres. This suggests that the growth is not linear because the increase is not the same each year. Instead, it suggests exponential growth, where the amount of kudzu increases by a certain percentage each year. Therefore, we can eliminate the linear options (C) and (D).Determining the Exponential Growth Rate: Now we need to determine the correct exponential growth rate. We can use the data point from 60 years after 1946, where the kudzu covered 7 million acres. We know that in 1946, it covered 3 million acres. The exponential growth function has the form K(t) = a*(1 + r)^t, where a is the initial amount, r is the growth rate, and t is the time in years. We can use the information given to solve for r.Solving for the Growth Rate: Let's plug in the values we know into the exponential growth formula. We have K(t) = 7 million acres when t = 60 years, and we know that the initial amount K(0) = 3 million acres. We need to find the rate r that satisfies the equation 7 = 3*(1 + r)^60.Comparing the Options: We can now compare the options given to see which one fits the form we need. Option (A) K(t)=3*(1.014)^t seems to be the correct form, as it represents an initial amount of 3 million acres growing at a certain rate over t years. Option (B) K(t)=3*(0.014)^t does not fit the form because it suggests the growth rate is 0.014, not 1.014, which would not result in exponential growth but rather exponential decay, which is not consistent with the information given.Confirming Option (A) is Correct: To confirm that option (A) is correct, we can check if it gives us the correct amount of kudzu in 60 years. Plugging t = 60 into the function K(t)=3*(1.014)^t, we get K(60) = 3*(1.014)^60. We need to calculate this value to see if it is approximately 7 million acres.Calculating K(60) = 3*(1.014)^60 using a calculator or a computer, we find that it is approximately 7 million acres (the exact value may vary slightly depending on the precision of the calculation). This confirms that option (A) is the correct model for the growth of kudzu.

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