Questions: Alana invests 1,000 into a continuously compounding account with an annual interest rate of 4 percent. Use the equation P(t)=1000 e^0.04 t to determine how much money will be in Alana's account after 15 years. (1 point) 1,173.51 1,630.97 1,822.12

Alana invests 1,000 into a continuously compounding account with an annual interest rate of 4 percent. Use the equation P(t)=1000 e^0.04 t to determine how much money will be in Alana's account after 15 years. (1 point) 1,173.51 1,630.97 1,822.12
Transcript text: Alana invests $\$ 1,000$ into a continuously compounding account with an annual interest rate of 4 percent. Use the equation $P(t)=1000 e^{0.04 t}$ to determine how much money will be in Alana's account after 15 years. (1 point) \$1, 173.51 \$1,630.97 \$1, 822.12
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Solution

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Solution Steps

Step 1: Identify the given values

The initial investment \( P_0 = \$1,000 \), the annual interest rate \( r = 0.04 \), and the time \( t = 15 \) years are given. The formula for continuously compounding interest is: \[ P(t) = P_0 e^{rt} \]

Step 2: Substitute the values into the formula

Substitute \( P_0 = 1000 \), \( r = 0.04 \), and \( t = 15 \) into the formula: \[ P(15) = 1000 e^{0.04 \cdot 15} \]

Step 3: Calculate the exponent

Calculate the exponent \( 0.04 \cdot 15 \): \[ 0.04 \cdot 15 = 0.6 \] So the equation becomes: \[ P(15) = 1000 e^{0.6} \]

Step 4: Evaluate \( e^{0.6} \)

Using a calculator, evaluate \( e^{0.6} \): \[ e^{0.6} \approx 1.8221 \]

Step 5: Calculate the final amount

Multiply \( 1000 \) by \( 1.8221 \): \[ P(15) = 1000 \cdot 1.8221 = 1822.1 \]

Final Answer

The amount in Alana's account after 15 years is: \[ \boxed{\$1,822.12} \]

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