t. Q (t)
0. 900
1. 802
2. 714
3. 636
4. 567
5. 505
6. 450
7. 401
8. 357
9. 318
10. 283
11. 253
12. 225
Construct an exponential decay functiom Q(t), where t is measured in minutes.
Q (t)=__?____ x (1/2)^ (_?_)
To construct an exponential decay function Q(t) using the given data, we can determine two key parameters: the initial value and the decay rate.
1. Initial Value (Q0):
The initial value represents the value of Q at t = 0. Looking at the data provided, we see that when t = 0, Q = 900. Therefore, the initial value (Q0) is 900.
2. Decay Rate (r):
The decay rate determines how quickly the value of Q decreases over time. In an exponential decay function, the general form is Q(t) = Q0 * (1/2)^(r*t), where r is the decay rate.
To find the decay rate (r), we need to find the value of the exponent that corresponds to a halving of the initial value. In other words, we need to determine the value of t for which Q(t) becomes exactly half of Q0.
From the given data, we can observe that when t = 1, Q = 802, and when t = 2, Q = 714. We can conclude that Q reduces by half between these two consecutive time points.
Using the formula for exponential decay, we can set up the equation:
Q(2) = Q(1) * (1/2)^(r*2)
Substituting the given values:
714 = 802 * (1/2)^(r*2)
To simplify the equation, let's divide both sides by 802:
714/802 = (1/2)^(2r)
Dividing 714 by 802 gives us approximately 0.8903. Therefore, the equation becomes:
0.8903 = (1/2)^(2r)
To solve for r, we can take the logarithm of both sides of the equation:
log(0.8903) = log((1/2)^(2r))
Using logarithmic properties, we can bring down the exponent:
log(0.8903) = 2r * log(1/2)
Now, divide both sides by 2 * log(1/2):
log(0.8903) / (2 * log(1/2)) = r
Evaluating this expression using a calculator, we find that r is approximately -0.1484.
Now that we have the initial value (Q0 = 900) and the decay rate (r = -0.1484), we can write the exponential decay function Q(t):
Q(t) = 900 * (1/2)^(t * -0.1484)
Simplifying further, we can write it as:
Q(t) = 900 * 2^(0.1484 * t)
Therefore, the exponential decay function Q(t) is:
Q(t) = 900 * 2^(0.1484 * t)