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15. Which of the following situations could be modeled by an exponential function? An 8 pound puppy gains 1.5 pounds per month for \( x \) number of months. The cost of an electrician that charges \( \$ 25 \) an hour for \( x \) number of hours. The amount in a savings account after \( \$ 75 \) is invested at a rate of \( 11 \% \) for \( x \) number of years. Theled by a car driving at 65 miles per hour for \( x \) number of hours. (1)

Ask by Montgomery Reeves. in the United States
Mar 14,2025

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Answer

The amount in a savings account after \$75 is invested at an 11% rate for \( x \) years can be modeled by an exponential function.

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To determine which of the given situations could be modeled by an exponential function, let's analyze each option step by step. 1. **An 8-pound puppy gains 1.5 pounds per month for \( x \) number of months.** - This situation describes a linear growth model because the puppy's weight increases by a constant amount (1.5 pounds) each month. The function can be expressed as: \[ W(x) = 8 + 1.5x \] - This is not an exponential function. 2. **The cost of an electrician that charges \( \$ 25 \) an hour for \( x \) number of hours.** - This situation also describes a linear relationship, as the cost increases by a constant rate of \( \$ 25 \) per hour. The function can be expressed as: \[ C(x) = 25x \] - This is not an exponential function. 3. **The amount in a savings account after \( \$ 75 \) is invested at a rate of \( 11\% \) for \( x \) number of years.** - This situation can be modeled by an exponential function because the amount in the savings account grows based on compound interest. The formula for compound interest is: \[ A(x) = P(1 + r)^x \] where \( P \) is the principal amount (\$75), \( r \) is the interest rate (0.11), and \( x \) is the number of years. Thus, it can be expressed as: \[ A(x) = 75(1 + 0.11)^x \] - This is an exponential function. 4. **The distance traveled by a car driving at 65 miles per hour for \( x \) number of hours.** - This situation describes a linear relationship, as the distance increases by a constant rate of 65 miles per hour. The function can be expressed as: \[ D(x) = 65x \] - This is not an exponential function. ### Conclusion: The only situation that could be modeled by an exponential function is: **The amount in a savings account after \( \$ 75 \) is invested at a rate of \( 11\% \) for \( x \) number of years.**

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The situation involving the savings account is a classic example of exponential growth. When money is invested at a certain interest rate, it doesn't just gain interest based on the initial amount; it also earns interest on the interest that has already been accumulated. This compound effect reflects the nature of exponential functions, showcasing how money can grow significantly over time, especially with a higher interest rate like 11%. On the other hand, the puppy's weight gain is modeled by a linear function, as it simply adds a fixed amount (1.5 pounds) each month. Similarly, the electrician's charge is linear too, operating on a steady hourly rate. These straightforward scenarios reflect how certain real-world situations can be effectively captured by linear models, contrasting with the more explosive growth seen with exponential functions. Choose wisely!

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