Asked by skungler
A company currently has 500 employees. The number of employees is expected to grow by 10% each year. Write an exponential function to model the number of employees in the company, `y`, after `x` years.
Answers
Answered by
GPT-4o mini
To model the growth of the number of employees in a company using an exponential function, we can use the formula:
\[
y = y_0 \cdot (1 + r)^x
\]
where:
- \( y \) is the number of employees after \( x \) years,
- \( y_0 \) is the initial number of employees,
- \( r \) is the growth rate (as a decimal),
- \( x \) is the number of years.
In your case:
- The initial number of employees (\( y_0 \)) is 500.
- The growth rate (\( r \)) is 10%, which is 0.10 in decimal form.
Plugging these values into the formula, we get:
\[
y = 500 \cdot (1 + 0.10)^x
\]
This simplifies to:
\[
y = 500 \cdot (1.10)^x
\]
Thus, the exponential function that models the number of employees in the company after \( x \) years is:
\[
y = 500 \cdot (1.10)^x
\]
\[
y = y_0 \cdot (1 + r)^x
\]
where:
- \( y \) is the number of employees after \( x \) years,
- \( y_0 \) is the initial number of employees,
- \( r \) is the growth rate (as a decimal),
- \( x \) is the number of years.
In your case:
- The initial number of employees (\( y_0 \)) is 500.
- The growth rate (\( r \)) is 10%, which is 0.10 in decimal form.
Plugging these values into the formula, we get:
\[
y = 500 \cdot (1 + 0.10)^x
\]
This simplifies to:
\[
y = 500 \cdot (1.10)^x
\]
Thus, the exponential function that models the number of employees in the company after \( x \) years is:
\[
y = 500 \cdot (1.10)^x
\]
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