16. The following rational function in hundreds models the population of a certain species of
animal, where t is measured in days. What number does the population approach in the long
run?
p(t)=10t^3+2/2t^3+1
17. The average cost of producing a popular board game is given by the function:
c(x)=1500+15x/x, , when x x>=0 is the number of the board game sold, identify the horizontal
asymptote of the function and explain its meaning in this context of this problem
I would really like help with these, and I know that they take a bit of time, but I really need help so that I can do well with calc.
animal, where t is measured in days. What number does the population approach in the long
run?
p(t)=10t^3+2/2t^3+1
17. The average cost of producing a popular board game is given by the function:
c(x)=1500+15x/x, , when x x>=0 is the number of the board game sold, identify the horizontal
asymptote of the function and explain its meaning in this context of this problem
I would really like help with these, and I know that they take a bit of time, but I really need help so that I can do well with calc.
-
A Rational Function is just a ratio of two polynomials:
f(x) = P(x) / Q(x)
To find the limit as x goes to infinity, you need to look at the degree of each polynomial (the highest exponent).
There are three possibilities for the limit of f(x) as x→∞:
1.) plus or minus infinity, if the degree of P(x) is more than the degree of Q(x)
2.) 0, if the degree of P(x) is less than the degree of Q(x)
3.) The ratio of the coefficients of the highest terms, if degree P(x) = degree Q(x)
So, for question 16
p(t) = (10t³ + 2) / (2t³ + 1)
So, the degree of the top and bottom are both the same (t to the 3rd power). So, you just need to take the ratio of the coefficients of the t³ terms.
lim p(t) = 10/2 = 5
t→∞
And since the model is in hundreds, that means, over the long run (as t goes to infinity) the population will approach 500.
For number 17:
c(x) = (1500 + 15x) / x
Again, the top and bottom have the same degree (x to the 1st power). So,
lim c(x) = 15/1 = 15
x→∞
That means as x gets bigger and bigger, c(x) will get closer and closer to 15. That's your horizontal asymptote. Or in the context of the problem, as you sell more and more board games, the cost of producing them will approach 15 (dollars?). So, maybe you should set the price at which you sell the board game with that in mind. If you wanted to make a certain percentage profit in the long run and expect to sell a lot, then this number is helpful. Especially, since if you only sell one board game, the cost of producing it is 1515 (dollars?). You obviously aren't going to sell it at that price.
I hope that helps!
f(x) = P(x) / Q(x)
To find the limit as x goes to infinity, you need to look at the degree of each polynomial (the highest exponent).
There are three possibilities for the limit of f(x) as x→∞:
1.) plus or minus infinity, if the degree of P(x) is more than the degree of Q(x)
2.) 0, if the degree of P(x) is less than the degree of Q(x)
3.) The ratio of the coefficients of the highest terms, if degree P(x) = degree Q(x)
So, for question 16
p(t) = (10t³ + 2) / (2t³ + 1)
So, the degree of the top and bottom are both the same (t to the 3rd power). So, you just need to take the ratio of the coefficients of the t³ terms.
lim p(t) = 10/2 = 5
t→∞
And since the model is in hundreds, that means, over the long run (as t goes to infinity) the population will approach 500.
For number 17:
c(x) = (1500 + 15x) / x
Again, the top and bottom have the same degree (x to the 1st power). So,
lim c(x) = 15/1 = 15
x→∞
That means as x gets bigger and bigger, c(x) will get closer and closer to 15. That's your horizontal asymptote. Or in the context of the problem, as you sell more and more board games, the cost of producing them will approach 15 (dollars?). So, maybe you should set the price at which you sell the board game with that in mind. If you wanted to make a certain percentage profit in the long run and expect to sell a lot, then this number is helpful. Especially, since if you only sell one board game, the cost of producing it is 1515 (dollars?). You obviously aren't going to sell it at that price.
I hope that helps!