Question

use the table to answer the question.

amount spend by ariel

week | 0 | 1 | 2 | 3 | 4
--- | --- | --- | --- | --- | ---
amount | $500 | $375 | $281 | $211 | ?

ariel received $500 for winning an art competition. she decided to limit her spending to 25 percent of the amount of money she has each week. the chart represents the amount of money ariel has left over after each week of spending. create an exponential formula to model the problem. how much would she have left after four weeks? round your answer to the nearest whole dollar.

(1 point)

rounded to the nearest whole dollar, ariel will have $\square$ after four weeks of spending.

Explanation:

Step1: Identify the exponential model

Ariel spends 25% of her money each week, so she keeps 100% - 25% = 75% of her money each week. The initial amount \( a = 500 \) dollars, and the common ratio \( r = 0.75 \) (since 75% = 0.75). The exponential formula for the amount of money left after \( t \) weeks is \( A(t)=a\times r^{t} \), where \( A(t) \) is the amount after \( t \) weeks, \( a \) is the initial amount, \( r \) is the ratio of the amount left each week, and \( t \) is the number of weeks.

Step2: Calculate the amount after 4 weeks

We need to find \( A(4) \) when \( a = 500 \), \( r = 0.75 \), and \( t = 4 \). Substitute these values into the formula:
\( A(4)=500\times(0.75)^{4} \)
First, calculate \( (0.75)^{4} \):
\( 0.75\times0.75 = 0.5625 \)
\( 0.5625\times0.75 = 0.421875 \)
\( 0.421875\times0.75 = 0.31640625 \)
Then, multiply by 500:
\( 500\times0.31640625 = 158.203125 \)

Step3: Round to the nearest whole dollar

Rounding 158.203125 to the nearest whole dollar gives 158.

Answer:

158