Question

Sally is modeling the change in diets among Native American populations around the Great Lakes by looking at the change over time of goosefoot seed remains in midden heaps. Midden heaps were locations where early peoples would dump the remains of food. She notices that the number of goosefoot seeds deposited in midden heaps has decreased by roughly \(7 \%\) per century, \(c\), since the earliest time period she studies. She estimates there were roughly 500 goosefoot seed remains deposited initially. Which of the following functions models \(S(c)\), the number of seeds found per century? A) \(S(c)=500(1.07)^c\) B) \(S(c)=500(0.93)^c\) C) \(S(c)=500^{0.93 c}\) D) \(S(c)=500^c\)

Short answer

The correct function that models the number of goosefoot seed remains per century is B) \(S(c)=500(0.93)^c\).

Step-by-step solution

Identify the initial number of seed remains

We are given that the initial number of seed remains is 500. This information will be used to eliminate options.

Find the correct growth factor

Since the number of seed remains decreases by 7% per century, we need to find a factor that represents this decrease. A decrease of 7% can be represented as a multiplication factor of 1 - 0.07 = 0.93. This means that the number of seed remains should be multiplied by 0.93 every century.

Analyze the given options

We will now analyze each option to see which one matches the information we have: A) \(S(c)=500(1.07)^c\): This represents an increase of 7% per century, not a decrease. Therefore, this is incorrect. B) \(S(c) = 500(0.93)^c\): This correctly represents the initial number of seed remains as 500 and shows a decrease of 7% per century. This is the correct option. C) \(S(c) = 500^{0.93c}\): This option has the correct growth factor (0.93). However, it does not represent an exponential decrease in the number of seed remains per century. D) \(S(c) = 500^c\): This option has neither the correct initial seed remains nor the correct growth factor. Thus, the correct answer is B) \(S(c)=500(0.93)^c\).