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A manufacturing plant earner $\$80$ per man-hour of labor when it opened. Each year, the plant earns an additional $5\%$ per man-hour. Write a function that gives the amount $A(t)$ that the plant earns per man-hour $t$ years after it opens.

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Write a linear function: word problems

Full solution

Q. A manufacturing plant earner

\$80

per man-hour of labor when it opened. Each year, the plant earns an additional

5\%

per man-hour. Write a function that gives the amount

A(t)

that the plant earns per man-hour

t

years after it opens.

Identify initial amount and increase: Identify the initial amount earned per man-hour and the annual percentage increase. $\newline$ The initial amount earned per man-hour is $\$80$ . The annual percentage increase is $5\%$ .
Convert percentage to decimal: Convert the annual percentage increase to a decimal. $\newline$ To convert a percentage to a decimal, divide by $100$ . Therefore, $5\%$ becomes $0.05$ .
Write function for earnings: Write the function for the amount earned per man-hour after $t$ years. $\newline$ The amount earned per man-hour after $t$ years can be represented by an exponential growth function because the earnings increase by a constant percentage each year. The general form of an exponential growth function is $A(t) = A_0 \times (1 + r)^t$ , where $A_0$ is the initial amount, $r$ is the growth rate, and $t$ is the time in years.
Substitute values into function: Substitute the known values into the exponential growth function. $\newline$ Using the initial amount $A_0 = \$80$ and the growth rate $r = 0.05$ , the function becomes $A(t) = 80 \times (1 + 0.05)^t$ .
Simplify the function: Simplify the function. $\newline$ The function simplifies to $A(t) = 80 \times (1.05)^t$ . This function represents the amount earned per man-hour $t$ years after the plant opens.

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