an initial investment of \$200 is now valued at \$350. the annual interest rate is 8\% compounded continuously. the equation $200e^{0.08t} = 350$ represents the situation, where $t$ is the number of years the money has been invested. about how long has the money been invested? use a calculator and round your answer to the nearest whole number.

5 years
7 years
19 years
22 years

Question

an initial investment of \$200 is now valued at \$350. the annual interest rate is 8\% compounded continuously. the equation $200e^{0.08t} = 350$ represents the situation, where $t$ is the number of years the money has been invested. about how long has the money been invested? use a calculator and round your answer to the nearest whole number.

5 years
7 years
19 years
22 years

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Exponential Equations" ], "new_concepts": [ "Logarithmic Equations" ], "current_concepts": [ "Exponential Equations", "Logarithmic Equations" ] } ## Isolate the exponential term $$ \begin{aligned} 200e^{0.08t} &= 350 \ e^{0.08t} &= \frac{350}{200} \ e^{0.08t} &= 1.75 \end{aligned} $$ ## Apply the natural logarithm $$ \begin{aligned} \ln(e^{0.08t}) &= \ln(1.75) \ 0.08t &= \ln(1.75) \end{aligned} $$ ## Solve for t and round $$ \begin{aligned} t &= \frac{\ln(1.75)}{0.08} \ t &\approx \frac{0.5596}{0.08} \ t &\approx 6.995 \approx 7 \end{aligned} $$

(A) 5 years

(B) 7 years

(C) 19 years

(D) 22 years

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Isolate the exponential term

Apply the natural logarithm

Solve for t and round

Answer: