Question
During a 5-year period of constant inflation, the value of a $104 comma 000 property increases according to the equation v equals 104 comma 000 e Superscript 0.04 t dollars. In how many years will the value of this building be double its current value?
Question content area bottom
Part 1
In
enter your response here years, the value of the building will double its current value.
(Type an integer or a decimal rounded to two decimal places as needed.)
Question content area bottom
Part 1
In
enter your response here years, the value of the building will double its current value.
(Type an integer or a decimal rounded to two decimal places as needed.)
Answers
GPT-4o mini
To find out when the value of the property will double, we first need to set up the equation for the doubled value:
The current value of the property is \( v = 104,000 \). When it doubles, the value becomes \( 2 \times 104,000 = 208,000 \).
According to the equation provided:
\[
v = 104,000 e^{0.04t}
\]
We want to set this equation equal to \( 208,000 \):
\[
208,000 = 104,000 e^{0.04t}
\]
Now, we can divide both sides by \( 104,000 \):
\[
2 = e^{0.04t}
\]
Next, we take the natural logarithm of both sides:
\[
\ln(2) = \ln(e^{0.04t})
\]
Using the property of logarithms that states \( \ln(e^x) = x \):
\[
\ln(2) = 0.04t
\]
Now, we solve for \( t \):
\[
t = \frac{\ln(2)}{0.04}
\]
Calculating \( \ln(2) \):
\[
\ln(2) \approx 0.6931
\]
Substituting this value in:
\[
t \approx \frac{0.6931}{0.04} \approx 17.3275
\]
Rounding this to two decimal places:
\[
t \approx 17.33
\]
Therefore, in approximately **17.33 years**, the value of the building will double its current value.
The current value of the property is \( v = 104,000 \). When it doubles, the value becomes \( 2 \times 104,000 = 208,000 \).
According to the equation provided:
\[
v = 104,000 e^{0.04t}
\]
We want to set this equation equal to \( 208,000 \):
\[
208,000 = 104,000 e^{0.04t}
\]
Now, we can divide both sides by \( 104,000 \):
\[
2 = e^{0.04t}
\]
Next, we take the natural logarithm of both sides:
\[
\ln(2) = \ln(e^{0.04t})
\]
Using the property of logarithms that states \( \ln(e^x) = x \):
\[
\ln(2) = 0.04t
\]
Now, we solve for \( t \):
\[
t = \frac{\ln(2)}{0.04}
\]
Calculating \( \ln(2) \):
\[
\ln(2) \approx 0.6931
\]
Substituting this value in:
\[
t \approx \frac{0.6931}{0.04} \approx 17.3275
\]
Rounding this to two decimal places:
\[
t \approx 17.33
\]
Therefore, in approximately **17.33 years**, the value of the building will double its current value.