The decay of a certain substance is modeled by the function N(t)=100(3+e^(-t/3), where the number of particles is represented by N in a time of t hours.
a) what is the initial number of particles?
- I set t as 0 and got an answer of 400 particles initially
b) determine the rate of change in the number of particles at time t
- Am I to find the derivative of the function? after that what would i do?
c) how fast is the number of particles changing when t=3 hours.
- would I sub in t=3 into the equation from part b?
a) correct
b) since you want the rate at a time of t, your derivative is your answer
c) you would sub t = 3 into the derivative, the rate of change, that you found in b)
thanks!
a) Correct! To find the initial number of particles, you substitute t = 0 into the function N(t). In this case, N(0) = 100(3 + e^0/3) = 100(3 + 1) = 400. So, the initial number of particles is indeed 400.
b) Yes, to determine the rate of change in the number of particles at time t, you need to find the derivative of the function N(t).
N'(t) = dN(t)/dt = d/dt (100(3 + e^(-t/3)))
To calculate this, you can use the chain rule. Differentiating 3 + e^(-t/3) with respect to t gives us 0 - (1/3)e^(-t/3). So, the derivative of N(t) with respect to t is:
N'(t) = 100 × (-1/3)e^(-t/3) = -100/3 × e^(-t/3)
This is the rate of change in the number of particles at time t.
c) To find how fast the number of particles is changing when t = 3 hours, you substitute t = 3 into the expression for the rate of change, N'(t).
N'(3) = -100/3 × e^(-3/3) = -100/3 × e^(-1)
Simplifying further, you'll get the exact value for the rate of change.
a) To find the initial number of particles, you can substitute t = 0 into the equation:
N(0) = 100(3 + e^(0/3))
N(0) = 100(3 + e^0)
N(0) = 100(3 + 1)
N(0) = 100(4)
N(0) = 400
So the initial number of particles is 400.
b) To determine the rate of change in the number of particles at time t, you need to take the derivative of the function N(t). The derivative will give you the rate of change, also known as the instantaneous rate of change or the slope of the function at any given point.
N(t) = 100(3 + e^(-t/3))
To differentiate N(t), you can use the chain rule, as the function contains both a constant (100) and the exponential function.
N'(t) = 100 * (0 + (-1/3)(e^(-t/3)))
N'(t) = -100/3 * e^(-t/3)
So the rate of change in the number of particles at time t is given by -100/3 times e raised to the power of -t/3.
c) To find how fast the number of particles is changing when t = 3 hours, you need to substitute t = 3 into the derivative function obtained in part b.
N'(3) = -100/3 * e^(-3/3)
N'(3) = -100/3 * e^(-1)
N'(3) ≈ -43.682
Therefore, when t = 3 hours, the rate of change in the number of particles is approximately -43.682 particles per hour.