24. A pharmaceutical company has determined that a realistic model for the relationship between blood pressure (systolic) and the amount of a nerve drug designed to reduce the systolic pressure is given by the equation P=Pi + 2x - x^2.

P is measured in millimeters of mercury and x is the number of milligrams of the drug administered 30 minutes before the reading of the pressure P. Pi is the systolic pressure before the medicine is taken and will vary with each patient. However, Pi must fall in the range 200 less than or equal to Pi greater than or equal to 220, for the model to be valid.

a. Determine the rate of change of the systolic pressure P with respect to the amount of drug, x, administered.

b. Determine the minimum amount of the drug that can be administered and still be effective.

c. If a particular patient has an initial systolic pressure of 200 and 10 milligrams of the drug administered, what will the pressure be 30 minutes later?

Note:
For part a. the answer is 2-2x. Is this because Pi is considered a number between 200 and 220 and the equation P=Pi + 2x - x^2 is then used to find the derivative.

a. dP/dx = 2 - 2x

b. In order for dP/dx to be negative (and the drug to therefore be effective, x must be >1 mg.
c. If x = 10 and Pi = 200, then P = Pi + 2x - 2x^2 after 30 minutes
= 200 +20 -200 = 20 mm mercury
Sounds like an overdose to me

Yes, you're correct in saying that the derivative of the function P = Pi + 2x - x^2 with respect to x gives us the rate of change of the systolic pressure P with respect to the amount of the drug x administered.

To find the derivative, we differentiate each term of the equation separately. The derivative of Pi with respect to x is 0 because Pi is a constant. The derivative of 2x with respect to x is 2, and the derivative of -x^2 with respect to x is -2x.

Therefore, the rate of change of the systolic pressure P with respect to x is given by the expression:

dP/dx = 2 - 2x

This shows that the rate of change of P decreases by 2 units for each milligram increase in the amount of the drug administered.

For part b, we want to determine the minimum amount of the drug that can be administered and still be effective. In this case, we need to find the minimum of the function P = Pi + 2x - x^2.

To find the minimum, we take the derivative with respect to x and set it equal to zero:

dP/dx = 2 - 2x = 0

Solving this equation, we find x = 1. This means that the minimum amount of the drug that can be administered and still be effective is 1 milligram.

For part c, if a particular patient has an initial systolic pressure of 200 and 10 milligrams of the drug administered, we can substitute these values into the equation P = Pi + 2x - x^2 to find the pressure 30 minutes later.

Given Pi = 200 and x = 10, we have:

P = 200 + 2(10) - (10)^2
P = 200 + 20 - 100
P = 120

Therefore, the systolic pressure 30 minutes later is 120 millimeters of mercury.