Tony's blood pressure can be modelled using the function P(t)=1000(sin t)(e^-2t), where 0 ≤ t ≤ 2 minutes. His body is stabilized after reaching a high. What is the highest blood pressure(nearest whole number) reached?
P = 1000 e^-2t sint
P' = 1000 e^-2t (cost - 2sint)
you need P'=0, so when is cost = 2sint in your domain?
I can't find t. When I tried, I got t=27, but when I graphed it, it show t=0.464
never mind, my calculator was in deg mode
Hi,
May I know how you got P' = 1000 e^-2t (cost - 2sint)?
I applied the product rule and I got P'(t)=1000(e^-2t cos(t) - 2e^-2t sin(t)), then I got stucked.
Can you please help me with this?
Thank you.
How do you solve for t?
To find the highest blood pressure reached by Tony, we need to find the maximum value of the function P(t) over the given time interval.
The function P(t) = 1000(sin t)(e^-2t) represents the blood pressure at time t. To find the maximum value, we can differentiate the function with respect to t and set it equal to zero.
Let's differentiate P(t) with respect to t:
P'(t) = 1000(cos t)(e^-2t) - 2000(sin t)(e^-2t)
Now, let's set P'(t) equal to zero and solve for t:
1000(cos t)(e^-2t) - 2000(sin t)(e^-2t) = 0
Dividing both sides by e^-2t, we get:
1000(cos t) - 2000(sin t) = 0
Dividing both sides by 1000, we get:
cos t - 2(sin t) = 0
Using the identity sin^2 t + cos^2 t = 1, we can rewrite the equation as:
1 - sin^2 t - 2sin t = 0
Rearranging the equation, we get:
sin^2 t + 2sin t - 1 = 0
Now, we can solve this quadratic equation for sin t. Let's use the quadratic formula:
sin t = (-2 ± √(2^2 - 4(1)(-1))) / (2)
sin t = (-2 ± √(4 + 4)) / 2
sin t = (-2 ± √8) / 2
sin t = (-2 ± 2√2) / 2
sin t = -1 ± √2
Since 0 ≤ t ≤ 2 minutes, we can discard the negative value for sin t. Therefore, sin t = √2.
Now, let's substitute sin t = √2 back into the original function P(t) to find the highest blood pressure:
P(t) = 1000(sin t)(e^-2t)
P(t) = 1000(√2)(e^-2t)
P(t) = 1000√2(e^-2t)
To find the highest blood pressure reached, we need to find the maximum value of e^-2t over the interval 0 ≤ t ≤ 2 minutes. Since e^-2t is a decreasing function, the highest value will be at t = 0.
Substituting t = 0 into e^-2t, we get:
e^-2(0) = e^0 = 1
Therefore, the highest blood pressure reached is:
P(t) = 1000√2(1) = 1000√2
Rounding to the nearest whole number, the highest blood pressure reached by Tony is approximately 1414.