Algebra 1 Question

Question

A biologist observes the growth of a deer population in a town. The biologist finds that the initial population of 75 deer triples every year.

Which equation can be used to find the number of years, x, it takes for the number of deer in the town to reach 1,125?

A. 1,125 = 3+75x

B. 1,125 = 3(75)x

C. 1,125 = 75+3x

D. 1,125 = 75(3)^x

Answer 1

This question asks us to create an exponential model of the form P=P₀(r)^t and then use substitution to find an equation that would let us solve for t.

Definition of the Equation P=P₀(r)^t

1. P is our population at any point in time, what we are trying to predict with our model
2. P₀ is our starting population, or the population at time t=0. The problem tells us this is 75
3. r is the rate of change in the population within a single time period. The problem tells us this is equal to 3 ("tripling")
4. t is our dependent variable, what we use to predict P. In this problem it is the number of years that have passed, or the variable x

Using this definition we get the model P=75(3)^x. We can then substitute the given population value to get the needed equation 1125=75(3)^x

Answer 2

If the population is tripling every 3 years, that's exponential growth (75, 225, 675, 2025, etc.). The only choice that has an exponent is D. You can try the others, but they won't give the results shown above.

For D, when x = 0, population is 3⁰ = 1 x 75 = 75. When x = 1 (one year later) population is 3¹ x 75 = 225. When x = 2, population is 3² = 9 x 75 = 675. And so on.

Also, I would argue that the x exponent as presented in choice D should be within the parentheses.

Answer 3

year zero 75(3)^0

year one 75(3)^1

year two 75(3)^2

year three 75(3)^3

year x 75(3)^x

1125 = 75(3)^x

3^x = 1125/75 = 15

3^x = 15

year two 3^2 = 9

year three 3^3 = 27

between 2 and 3 years the population = 1125

3^x = 15

x = log₃15 = ln15/ln3 = about 2.465 years

= 2 years and 170 days

D) is the correct answer