Select the correct answer.The population of snakes at a wildlife park over the course of six months is modeled by the equation y = x5 − 5x4 + 2x3 − 10x2 + 100x + 5, where x is the number of months. If the park officials count 505 snakes, how many months have passed since the initial count was taken? A. 3 months B. 4 months C. 5 months D. 6 months
Question
Select the correct answer.The population of snakes at a wildlife park over the course of six months is modeled by the equation y = x5 − 5x4 + 2x3 − 10x2 + 100x + 5, where x is the number of months. If the park officials count 505 snakes, how many months have passed since the initial count was taken? A. 3 months B. 4 months C. 5 months D. 6 months
Solution
To solve this problem, we need to substitute the given population of snakes (y = 505) into the equation and solve for x (the number of months).
The equation is: y = x^5 - 5x^4 + 2x^3 - 10x^2 + 100x + 5
Substitute y = 505 into the equation:
505 = x^5 - 5x^4 + 2x^3 - 10x^2 + 100x + 5
This equation is a quintic equation (a polynomial of degree 5), which can be difficult to solve analytically. However, since the problem provides us with multiple choices, we can substitute each option into the equation and see which one gives us 505.
Let's start with option A (x = 3):
505 = 3^5 - 53^4 + 23^3 - 103^2 + 1003 + 5 = 243 - 405 + 54 - 90 + 300 + 5 = 107
Option A does not give us 505, so it's not the correct answer.
Let's try option B (x = 4):
505 = 4^5 - 54^4 + 24^3 - 104^2 + 1004 + 5 = 1024 - 1280 + 128 - 160 + 400 + 5 = 117
Option B also does not give us 505, so it's not the correct answer.
Let's try option C (x = 5):
505 = 5^5 - 55^4 + 25^3 - 105^2 + 1005 + 5 = 3125 - 3125 + 250 - 250 + 500 + 5 = 505
Option C gives us 505, so it's the correct answer.
Therefore, 5 months have passed since the initial count was taken.
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