Question

1. assignment: exponential growth and decay

which of the following scenarios does the function $f(x) = 288(0.84)^x$ best describe?
a. a gift card with an initial balance of $288 that decreases at a rate of 84% each month.
a. a gift card with an initial balance of $288 that decreases by $0.84 each month
a. a gift card with an initial balance of $0.84 that increases at a rate of 288% each month
a. a gift card with an initial balance of $288 that decreases at a rate of 16% each year.

Explanation:

Step1: Recall Exponential Decay Formula

The general form of an exponential decay function is \( f(x) = a(1 - r)^x \), where \( a \) is the initial amount, \( r \) is the rate of decay (as a decimal), and \( x \) is the independent variable. In the given function \( f(x)=288(0.84)^x \), we can compare it to this form. Here, \( a = 288 \) (initial balance) and \( 1 - r=0.84 \).

Step2: Calculate the Decay Rate

To find \( r \), we solve \( 1 - r = 0.84 \). Subtracting 0.84 from 1 gives \( r=1 - 0.84 = 0.16 \), which is 16% when converted to a percentage. But wait, let's check the options again. Wait, the base is 0.84, so the remaining amount each time is 84% of the previous, meaning the decay rate is \( 1 - 0.84=0.16 \) or 16%? Wait no, wait the options: let's analyze each option. Option D: "A gift card with an initial balance of $288 that decreases at a rate of 16% each year." Wait, but let's check the base. Wait, the function is \( f(x)=288(0.84)^x \). So the multiplier is 0.84, which means each time \( x \) increases by 1, the value is 84% of the previous, so the decay rate is \( 1 - 0.84 = 0.16 \) or 16%. Wait, but let's check the options again. Wait the options:

• Option A: decreases at 84% each month? No, 0.84 is the multiplier, not the rate.
• Option B: increases? No, 0.84 < 1, so it's decay.
• Option C: decreases by $0.84 each month? That's linear decay, not exponential.
• Option D: decreases at 16% each year (or whatever x is, but the initial balance is 288, and the multiplier is 0.84, so 1 - 0.84 = 0.16 = 16% decay rate. So the function \( f(x)=288(0.84)^x \) represents a gift card with initial balance $288 that decreases at a rate of 16% (since 1 - 0.84 = 0.16) each time x increases. So the correct option is D. Wait, but let's confirm again. The initial balance is 288, the base is 0.84, so each time, the balance is 84% of the previous, so it's decreasing by 16% (100% - 84% = 16%) each period (x). So the correct option is D: "A gift card with an initial balance of $288 that decreases at a rate of 16% each year." Wait, but the options: let's check the original image. Wait the options are:

A. A gift card with an initial balance of $288 that decreases at a rate of 84% each month.

B. A gift card with an initial balance of $0.84 that increases at a rate of 288% each month.

C. A gift card with an initial balance of $288 that decreases by $0.84 each month.

D. A gift card with an initial balance of $288 that decreases at a rate of 16% each year.

Wait, so the correct analysis: the function is exponential decay, initial amount 288, base 0.84 (so 84% remaining, 16% decay). So the correct option is D.

Answer:

D. A gift card with an initial balance of $288 that decreases at a rate of 16% each year.