Question

question 3 adam used the function $c(x)=50(0.75)^x$ to model the balance in his checking account $x$ days after he deposited his birthday money. which statement is the best interpretation of one of the values in this function? a adam’s account balance decreases at a rate of 75% each day. b adam’s account balance increases at a rate of 25% each day. c the amount in adam’s account after one day is $50. d adam deposited $50 into his checking account.

Explanation:

Step1: Recall the exponential function form

The general form of an exponential function is \( c(x)=a(b)^x \), where \( a \) is the initial value, \( b \) is the growth/decay factor. If \( 0 < b < 1 \), it's a decay function, and the decay rate is \( 1 - b \).

Step2: Analyze the given function \( c(x)=50(0.75)^x \)

• For option A: The decay factor is \( 0.75 \), so the decay rate is \( 1 - 0.75 = 0.25 = 25\% \) per day, not 75%. So A is wrong.
• For option B: Since \( 0.75<1 \), the function represents decay (decrease), not increase. So B is wrong.
• For option C: When \( x = 1 \), \( c(1)=50(0.75)^1 = 37.5

eq50 \). So C is wrong.

• For option D: When \( x = 0 \) (initial time, 0 days after deposit), \( c(0)=50(0.75)^0 = 50(1)=50 \). This means the initial deposit (when \( x = 0 \)) is $50. So D is correct.

Answer:

D. Adam deposited $50 into his checking account.