a. To find the blood pressure after 30 seconds, we substitute t = 30 into the equation P = 20sin(2πt) + 90 and evaluate:
P = 20sin(2π(30)) + 90
P = 20sin(60π) + 90
P = 20(0) + 90 (since sin(60π) = 0)
P = 90
Therefore, the blood pressure after 30 seconds is 90.
b. The maximum blood pressure occurs when sin(2πt) = 1, which happens when 2πt = π/2 + 2πn for some integer n. Solving for t, we get:
2πt = π/2 + 2πn
t = (π/4 + πn)/π = 1/4 + n
Since t represents time in seconds, the values of t that satisfy the equation must be non-negative and less than or equal to 60 (since the blood pressure is modeled for one minute or 60 seconds). Therefore, we get:
t = 1/4, 1/4 + 1, 1/4 + 2, ..., 1/4 + 239
We can see that the maximum value of P occurs when t = 1/4 + 120 = 120.25 seconds (the middle of the time interval), and we can evaluate it using the equation:
P = 20sin(2π(120.25)) + 90
P ≈ 110.9
Therefore, the maximum blood pressure is approximately 110.9.
The minimum blood pressure occurs when sin(2πt) = -1, which happens when 2πt = 3π/2 + 2πn for some integer n. Solving for t, we get:
2πt = 3π/2 + 2πn
t = (3π/4 + πn)/π = 3/4 + n
Again, since t represents time in seconds, the values of t that satisfy the equation must be non-negative and less than or equal to 60. Therefore, we get:
t = 3/4, 3/4 + 1, 3/4 + 2, ..., 3/4 + 79
We can see that the minimum value of P occurs when t = 3/4 + 39 = 39.75 seconds (the middle of the time interval), and we can evaluate it using the equation:
P = 20sin(2π(39.75)) + 90
P ≈ 69.1
Therefore, the minimum blood pressure is approximately 69.1.