1. The population of an endangered bird is decreasing at a rate of 0.75%
per year. There are currently about 200,000 of these birds.
a. What exponential function would be a good model for the population of
these endangered birds?
b. How many birds will there be in 100 years?
2. How much money invested at 6% compounded continously for 4 years
will result in $2500?
3. Write the equation in logarithm form 8^3 = 512
4. Evaluate the logarithm log3 1/81
1. a. The exponential function that models the population of these endangered birds can be written as: P(t) = P0 * (1 - r)^t, where P(t) is the population at time t, P0 is the initial population, r is the rate of decrease per year, and t is the number of years.
In this case, P0 = 200,000 and r = 0.75% = 0.0075. Therefore, the exponential function is:
P(t) = 200,000 * (1 - 0.0075)^t.
b. To find the population of birds in 100 years, we can substitute t = 100 into the exponential function:
P(100) = 200,000 * (1 - 0.0075)^100.
2. The formula for continuous compound interest is: A = P * e^(rt), where A is the final amount, P is the principal amount, r is the annual interest rate (in decimal form), t is the time in years, and e is the base of the natural logarithm.
In this case, we want to find the principal amount. Let P be the unknown principal amount.
$2500 = P * e^(0.06 * 4).
To solve for P, we divide both sides by e^(0.06 * 4):
P = $2500 / e^(0.24).
3. In logarithm form, the equation 8^3 = 512 can be written as log8 512 = 3. This means that the logarithm base 8 of 512 is equal to 3.
4. To evaluate log3 (1/81), we need to find the exponent to which the base 3 must be raised to obtain 1/81.
3^x = 1/81
Rewrite 1/81 as 3^(-4):
3^x = 3^(-4)
Since the bases are the same, the exponents must be equal:
x = -4.
Therefore, log3 (1/81) = -4.