find an exponential function of the form P(t)yoe^kt to model the given;
In 1995, the number of female athletes participating in Summer Olympics-Type Games was 550. In 1996, about 3550 participated in the summer olympics in ATlanta. Assuming that P(0)= 500 and that the exponential model applied, find the value of k rounded to the hundredths place, and write the function
What is yo? What year (y) is 1995? Does the 3550 total apply to women or both sexes? How do "Olympics-Type Games" in 1995 compare to real Olympics in 1996?
It is hard to fit a model without knowing what the terms mean
To find the exponential function to model the given scenario, we'll use the formula: P(t) = P(0) * e^(kt), where P(t) represents the number of female athletes at year t, P(0) represents the initial number of female athletes, e is Euler's number (approximately 2.71828), and k is the constant we need to determine.
Given information:
P(0) = 500 (number of female athletes in 1995)
P(1) = 3550 (number of female athletes in 1996)
We can use these two points to form two equations:
Equation 1: P(0) = 500 = P(0) * e^(k*0)
Equation 2: P(1) = 3550 = P(0) * e^(k*1)
Simplifying Equation 1:
500 = 500 * e^0
Since any value raised to the power of 0 is 1, we have:
500 = 500 * 1
Therefore, Equation 1 is satisfied.
Simplifying Equation 2:
3550 = 500 * e^k
Now, to find the value of k, we'll divide both sides of Equation 2 by 500:
3550/500 = e^k
Simplifying further:
7.1 = e^k
To find the value of k, we can take the natural logarithm (ln) of both sides:
ln(7.1) = ln(e^k)
ln(7.1) = k
Hence, rounded to the hundredths place, the value of k is approximately 1.96.
Now that we have the value of k, we can write the exponential function:
P(t) = 500 * e^(1.96t)
Therefore, the exponential function to model the given scenario is P(t) = 500 * e^(1.96t).