Question

1. a new animal species is introduced to an uninhabited island. the species has an abundant food source and its population thrives. sketch a graph showing what the population of the species might look like over time.

Explanation:

Problem 5:

Step1: Define variables

Let \( q \) be the number of quarters and \( n \) be the number of nickels.
We know two equations:

1. Total coins: \( q + n = 13 \)
2. Total value: \( 0.25q + 0.05n = 1.25 \) (since a quarter is $0.25 and a nickel is $0.05)

Step2: Solve the system of equations

From the first equation, we can express \( n = 13 - q \).
Substitute \( n = 13 - q \) into the second equation:
\( 0.25q + 0.05(13 - q) = 1.25 \)
Expand: \( 0.25q + 0.65 - 0.05q = 1.25 \)
Combine like terms: \( 0.2q + 0.65 = 1.25 \)
Subtract 0.65 from both sides: \( 0.2q = 1.25 - 0.65 = 0.6 \)
Divide by 0.2: \( q = \frac{0.6}{0.2} = 3 \)

Step3: Find the number of nickels

Substitute \( q = 3 \) into \( n = 13 - q \): \( n = 13 - 3 = 10 \)

The population of the animal species, with abundant food and no initial competition/predators, follows exponential growth initially. The graph should have time (x - axis) and population (y - axis). It starts at a small initial population (since a new species is introduced) and increases rapidly, curving upwards (exponential growth curve, \( P(t)=P_0e^{rt} \) where \( P_0 \) is initial population, \( r \) is growth rate, \( t \) is time). As time increases, the population grows faster (steeper slope as \( t \) increases) because the growth is proportional to the current population.

Answer:

Number of quarters: 3, Number of nickels: 10

Problem 6: