Question

question 20 of 25
the price of 9 - volt batteries is increasing according to the function below,
where ( t ) is years after january 1, 1980. during what year will the price reach
( $ 3 )?
( p(t)=1.1 cdot e^{0.047 t} )
a. 2001
b. 1993
c. 1997
d. 2005

Explanation:

Step1: Set up the equation

Set \(P(t)=3\), so \(3 = 1.1\times e^{0.047t}\).
Then \(\frac{3}{1.1}=e^{0.047t}\), which simplifies to \(\frac{30}{11}=e^{0.047t}\).

Step2: Take the natural logarithm

Take the natural logarithm of both sides: \(\ln(\frac{30}{11})=\ln(e^{0.047t})\).
Since \(\ln(e^{x}) = x\), we have \(\ln(\frac{30}{11}) = 0.047t\).
Calculate \(\ln(\frac{30}{11})\approx1.003\).
So \(t=\frac{1.003}{0.047}\approx21.34\).

Step3: Calculate the year

Since \(t\) is years after 1980, the year is \(1980 + 21=2001\) (rounding \(t\approx21\)).

Answer:

A. 2001