Question

Exponential functions. A population of Alaskan king crabs is being monitored. Initially, there are 23 crabs, and the population grows by 8% each year.

Classify as: growth or decay
Create the function:
Domain:
Range:
y-intercept:

A) Growth, P(t) = 23(1.08)^t, Domain: t >= 0, Range: P > 23, y-intercept: (0, 23)
B) Decay, P(t) = 23(0.08)^t, Domain: t >= 0, Range: P < 23, y-intercept: (0, 23)
C) Growth, P(t) = 23(0.08)^t, Domain: t >= 0, Range: P > 23, y-intercept: (0, 0)
D) Decay, P(t) = 23(1.08)^t, Domain: t >= 0, Range: P < 23, y-intercept: (0, 0)

Answer 1

The correct option is A) Growth, with the exponential growth function P(t) = 23(1.08)^t. This represents an Alaskan king crab population increasing by 8% annually, with Domain: t ≥ 0, Range: P > 23, and a y-intercept of (0, 23).

Step-by-step explanation:

The question pertains to the exponential growth of a population of Alaskan king crabs. Since the population is increasing by 8% each year, this is an example of growth, not decay. We must look for a function where the base of the exponent is greater than 1, which indicates growth, as opposed to a base less than 1 that would indicate decay. The initial population is 23 crabs, and the growth rate is 8%. The correct exponential function would be of the form P(t) = P0ert, where P0 is the initial population, r is the growth rate, and t is time in years. In this case, the rate is expressed as a decimal, so an 8% growth rate is 0.08, and the function becomes:

P(t) = 23(1.08)^t

We can then classify the components of the function as follows:

• Domain: The domain is all non-negative numbers, representing time, so t ≥ 0.
• Range: The range is all population sizes greater than the original 23, so P > 23.
• y-intercept: The y-intercept is the initial population size when t = 0, which is (0, 23).

Therefore, the correct option is A) Growth, P(t) = 23(1.08)^t, Domain: t ≥ 0, Range: P > 23, y-intercept: (0, 23).