The function f(x)=350(0.75)^x models the amount of money Alicia has in her bank account last summer, where x represents the number of weeks since summer began.

Question

1a. is the money in her account growing or decaying?
1b. at what rate does is it growing or decaying?.
1c. what does the 350 represent?
2. how much money does alicia have left after 2 weeks of summer?. round your answer to the nearest whole number

Answer 1

1a. The money in Alicia's account is decaying. This is indicated by the base of the exponential function, which is \(0.75\). Since it is less than 1, it shows that the amount of money decreases over time.

1b. The money is decaying at a rate of \(25%\). This is calculated by using the formula for exponential decay, where the rate of decay can be found by subtracting the base from 1: \(1 - 0.75 = 0.25\), which means it is losing \(25%\) of its value each week.

1c. The value \(350\) represents the initial amount of money in Alicia's bank account at the start of the period (when \(x = 0\)), which is the amount she had last summer.

2. To find out how much money Alicia has left after 2 weeks of summer, we can substitute \(x = 2\) into the function \(f(x) = 350(0.75)^x\):

\[ f(2) = 350(0.75)^2 \]

Now, let's calculate \(0.75^2\):

\[ 0.75^2 = 0.5625 \]

Now, substituting this back into the function gives:

\[ f(2) = 350 \times 0.5625 = 196.875 \]

Rounding this to the nearest whole number, Alicia has approximately \(197\) dollars left after 2 weeks of summer.