An investment lost approximately $5 \%$ of the balance each month for the past year. The amount of the investment on January $1^{\text {st }}$ of last year was $\$ 10,000$ . Which of the following functions, $I$ , models the amount of the investment (in thousands of dollars) at the end of month, $n$ , where $1 \leq n \leq 12$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $I(n)=10 \cdot(1.05)^{n}$ thousand dollars $\newline$ (B) $I(n)=10-0.05 n$ thousand dollars $\newline$ (C) $I(n)=10 \cdot(0.95)^{n}$ thousand dollars $\newline$ (D) $I(n)=10+0.95 n$ thousand dollars

View step-by-step help

Home

Math Problems

Algebra 1

Compare linear and exponential growth

Full solution

Q. An investment lost approximately

5 \%

of the balance each month for the past year. The amount of the investment on January

1^{\text {st }}

of last year was

\$ 10,000

. Which of the following functions,

I

, models the amount of the investment (in thousands of dollars) at the end of month,

n

, where

1 \leq n \leq 12

\newline

Choose

1

answer:

\newline

(A)

I(n)=10 \cdot(1.05)^{n}

thousand dollars

\newline

(B)

I(n)=10-0.05 n

thousand dollars

\newline

(C)

I(n)=10 \cdot(0.95)^{n}

thousand dollars

\newline

(D)

I(n)=10+0.95 n

thousand dollars

Understand the problem: Understand the problem. $\newline$ The investment loses $5\%$ of its balance each month. This means that each month, the investment retains $95\%$ ( $100\% - 5\%$ ) of its value from the previous month. We need to find a function that models this situation over a period of $12$ months.
Analyze the answer choices: Analyze the answer choices. $\newline$ We need to find a function that decreases the investment by $5\%$ each month. This means we are looking for an exponential decay function because the investment is losing a constant percentage, not a constant amount.
Evaluate the answer choices: Evaluate the answer choices. $\newline$ (A) $I(n)=10\times(1.05)^{n}$ thousand dollars - This function suggests the investment is increasing by $5\%$ each month, which is incorrect. $\newline$ (B) $I(n)=10-0.05n$ thousand dollars - This function suggests the investment is decreasing by a constant amount each month, which is incorrect. $\newline$ (C) $I(n)=10\times(0.95)^{n}$ thousand dollars - This function suggests the investment is decreasing by $5\%$ each month, which is correct. $\newline$ (D) $I(n)=10+0.95n$ thousand dollars - This function suggests the investment is increasing by a constant amount each month, which is incorrect.
Choose the correct function: Choose the correct function. $\newline$ The correct function is the one that models a $5\%$ decrease each month. The only function that represents a $5\%$ decrease each month is option (C) $I(n)=10\times(0.95)^n$ thousand dollars.