3. To estimate animal populations, biologists count the total number of animals in a small section of a habitat. The total population of animals is directly proportional to the size of the habitat (in acres) polled.
a. Write an equation using only one variable that could be used to solve for the constant of variation k.
b. A biologist counted 12 white tail deer in a 100-acre parcel of land in a nature preserve. Find the constant of variation k.
c. If the entire nature preserve is 2,500 acres, then what is the total white tail deer population in the preserve? Describe how you arrived at your answer.
a. To write an equation using only one variable to solve for the constant of variation k, we can let P represent the population and H represent the size of the habitat in acres. Since the total population is directly proportional to the size of the habitat, the equation can be written as:
P = kH
Here, k is the constant of variation.
b. We are given that the biologist counted 12 white-tail deer in a 100-acre parcel of land. Using the equation from part a, we can plug in the values:
12 = k(100)
To solve for k, divide both sides of the equation by 100:
12/100 = k
Simplifying, we get:
k = 0.12
So, the constant of variation is 0.12.
c. Now, we need to find the total white-tail deer population in the 2,500-acre nature preserve. We can use the equation from part a, and substitute the known values:
P = kH
P = 0.12(2,500)
Simplifying, we get:
P = 300
Therefore, the total white-tail deer population in the preserve is 300. To arrive at this answer, we used the constant of variation k obtained in part b and substituted it into the equation along with the size of the habitat (2,500 acres).