Question

A population of deer is increasing exponentially. Which exponential equation in the form y=abx
represents the situation shown in the graph?



(1 point)
Responses

y=90(3)x
y is equal to 90 times 3 to the x th power

y=60(32)x
y is equal to 60 times 3 halves to the x th power

y=90(52)x
y is equal to 90 times 5 halves to the x th power

y=60(23)x
y is equal to 60 times 2 thirds to the x th power

Answer 1

To determine the correct exponential equation representing the population of deer increasing in the form \( y = ab^x \), we need to analyze the characteristics of the options given.

In the equation \( y = ab^x \):

- \( a \) represents the initial amount (the population at \( x = 0 \)).
- \( b \) represents the growth factor. If \( b > 1 \), the population is increasing; if \( b < 1 \), the population is decreasing.

The equations provided are:

1. \( y = 90(3)^x \)
2. \( y = 60(3/2)^x \)
3. \( y = 90(5/2)^x \)
4. \( y = 60(2/3)^x \)

Options 1, 2, and 3 have a growth factor \( b > 1 \), indicating that the population is increasing:

1. **Option 1**: \( 90(3)^x \) - Growth factor \( b = 3 \) (very high increase).
2. **Option 2**: \( 60(3/2)^x \) - Growth factor \( b = 1.5 \) (increasing).
3. **Option 3**: \( 90(5/2)^x \) - Growth factor \( b = 2.5 \) (increasing).
4. **Option 4**: \( 60(2/3)^x \) - Growth factor \( b = 0.67 \) (decreasing).

Assuming the population of deer is indeed increasing, we can disregard Option 4. Now, which option is best?

To choose between options 1, 2, and 3, we need to consider the initial population:

- If the initial size of the population is around 90, Options 1 and 3 would be valid.
- If the initial size is around 60, then Option 2 would be valid.

Without additional context or a graph, any of the first three could work based on the assumption of initial population and growth rate. However, if we assume that populations typically grow more gradually in natural contexts (and without more details), Option 2 \( y = 60(3/2)^x \) might represent a reasonable balance between growth and initial population.

Choose **Option 2: \( y = 60(3/2)^x \)** as a reasonable solution to depict a sustainably growing deer population, provided that the initial population aligns with 60.