The concentration of a medicine, f(x) in parts per million, in a patient's blood X hours after the medicine is taken is given by the function

f(x)= -x^4 +12x^3 -58x^2 +132x.

1.) How many hours after the medicine is taken will it be eliminated from the bloodstream?

2.) What will be the level of medicine in the body after 1, 2, 3, 4, and 5 hours?
For question 2, do I just plug in 1, 2, 3, 4, and 5 separately into the function given above and solve for each so I would get 5 answers??

Yes

Did you get #1 ?

1.) To find the time when the medicine will be eliminated from the bloodstream, we need to determine when the concentration, f(x), will be equal to zero.

Setting f(x) = 0:
- x^4 + 12x^3 - 58x^2 + 132x = 0

To solve this equation, we can factor out an "x":
x(x^3 + 12x^2 - 58x + 132) = 0

Then, we can factor further using synthetic division or other methods to find the roots. However, in this case, we can observe that x = 0 is one of the roots.

So, one hour after the medicine is taken, it will be eliminated from the bloodstream.

2.) Yes, to determine the concentration of the medicine in the body after 1, 2, 3, 4, and 5 hours, you simply plug in each value separately into the function f(x) = -x^4 + 12x^3 - 58x^2 + 132x and solve for each.

For example:
- After 1 hour: f(1) = -(1)^4 + 12(1)^3 - 58(1)^2 + 132(1) = -1 + 12 - 58 + 132 = 85 ppm
- After 2 hours: f(2) = -(2)^4 + 12(2)^3 - 58(2)^2 + 132(2) = -16 + 96 - 232 + 264 = 112 ppm

You can repeat this process for 3, 4, and 5 hours to find the concentration of the medicine at each time point.

1.) To find the time when the medicine is eliminated from the bloodstream, we need to find the value of x for which f(x) is equal to zero. This means solving the equation f(x) = 0.

We are given the function f(x) = -x^4 + 12x^3 - 58x^2 + 132x. So we need to solve the equation:

-x^4 + 12x^3 - 58x^2 + 132x = 0.

To solve this equation, you can try factoring, but in this case, factoring might not be straightforward. So an alternative approach would be to use numerical methods, such as graphing the function or using a calculator or software that can find the roots, or zeros, of a function.

2.) To find the level of medicine in the body after 1, 2, 3, 4, and 5 hours, you can indeed plug in each value separately into the function f(x):

For x = 1, f(1) = -1^4 + 12(1)^3 - 58(1)^2 + 132(1) = -1 + 12 - 58 + 132 = 85.
So after 1 hour, the level of medicine in the body is 85 parts per million.

Similarly, for x = 2, f(2) = -2^4 + 12(2)^3 - 58(2)^2 + 132(2) = -16 + 96 - 232 + 264 = 112.
So after 2 hours, the level of medicine in the body is 112 parts per million.

Repeat this process for x = 3, 4, and 5, plugging in each value into the function and performing the calculations to find the level of medicine in the body at each respective time.

Alternatively, you can create a table and list the values of x (the time in hours) in one column and the corresponding values of f(x) (the medicine concentration) in the second column. This will allow you to organize the calculations and record the values more efficiently.