Question

Heather just got hired as an administrative assistant at Haven Enterprises. Her starting salary is $45,500, and her contract ensures that she will get a 3% salary increase each year.
Write an exponential equation in the form y=a(b)x that can model Heather's salary, y, after x years.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
Answer
When Leslie first moved to Lime County one year ago, it had a population of about 711,500. Today, the population in the county is about 697,270, and it is expected that the population will continue to decline each year.
Write an exponential equation in the form y=a(b)^x that can model the population of Lime County, y, x years after Leslie moved there.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =
Answer
Use the following function rule to find f(1).

f(x) =
–6
( 10 )
x


f(1) =
Answer
Use the following function rule to find f(2).

f(x) =
7.9
( 10 )
x


Write your answer as a decimal or whole number.

f(2) =
Answered by GPT-4o mini
To model Heather's salary over time, we can use the exponential equation in the form \( y = a(b)^x \).

Here, we need to identify the values for \( a \) and \( b \):

- \( a \) is the initial salary, which is \( 45,500 \).
- \( b \) is the growth factor. Since Heather receives a 3% increase each year, her salary will be multiplied by \( 1 + 0.03 = 1.03 \).

Therefore, the equation modeling Heather's salary after \( x \) years is:

\[
y = 45500(1.03)^x
\]
Answered by GPT-4o mini
To model the declining population of Lime County, we use the exponential equation in the form \( y = a(b)^x \).

Here, we need to identify the values for \( a \) and \( b \):

- **\( a \)** is the initial population when Leslie moved there, which is \( 711,500 \).

- **\( b \)** is the decay factor. To find \( b \), we first calculate the rate of decline. The population declined from \( 711,500 \) to \( 697,270 \) over one year.

To find the decay rate:
\[
\text{Decline} = 711500 - 697270 = 1230
\]

Next, we calculate the new population as a fraction of the original:
\[
b = \frac{\text{New Population}}{\text{Initial Population}} = \frac{697270}{711500} \approx 0.98094
\]

Thus, rounding to an appropriate decimal, we can use \( b \approx 0.981 \).

Putting this together, the exponential equation modeling the population after \( x \) years is:

\[
y = 711500(0.981)^x
\]

So, the final equation is:

\[
y = 711500(0.981)^x
\]
Answered by GPT-4o mini
To find \( f(1) \) using the given function rule \( f(x) = -6(10)^x \), we can substitute \( x = 1 \) into the equation.

\[
f(1) = -6(10)^1
\]

Now calculate \( (10)^1 \):

\[
f(1) = -6 \cdot 10
\]

Now carry out the multiplication:

\[
f(1) = -60
\]

Thus, the result is:

\[
f(1) = -60
\]
Answered by GPT-4o mini
To find \( f(2) \) using the given function rule \( f(x) = 7.9(10)^x \), we substitute \( x = 2 \) into the equation:

\[
f(2) = 7.9(10)^2
\]

Now calculate \( (10)^2 \):

\[
(10)^2 = 100
\]

Now substitute back into the equation:

\[
f(2) = 7.9 \cdot 100
\]

Now perform the multiplication:

\[
f(2) = 790
\]

Thus, the result is:

\[
f(2) = 790
\]