Question

6.4 exponential growth and decay quiz

which equation represents exponential growth of \\$1,200 at a rate of 4\\% per year?

\\(y = 1200(0.96)^t\\)
\\(y = 1200(1.04)^t\\)
\\(y = 1200 + 0.04t\\)
\\(y = 1200t^4\\)

Explanation:

Identify the exponential growth formula

The general formula for exponential growth is:

$$y = a(1 + r)^t$$

where:

• \(a\) is the initial amount.
• \(r\) is the growth rate (as a decimal).
• \(t\) is the time (usually in years).
• \((1 + r)\) is the growth factor.

Extract the given values

From the problem description, we have:

• Initial amount \(a = 1200\)
• Growth rate \(r = 4\% = 0.04\) per year

Calculate the growth factor

Substitute the rate \(r = 0.04\) into the growth factor expression:

$$1 + r = 1 + 0.04 = 1.04$$

Substitute values into the formula

Substitute \(a = 1200\) and the growth factor \(1.04\) into the general equation:

$$y = 1200(1.04)^t$$

Match with the given options

Compare our derived equation with the choices:

• \(y = 1200(0.96)^t\) represents decay.
• \(y = 1200(1.04)^t\) represents growth.
• \(y = 1200 + 0.04t\) is a linear equation.
• \(y = 1200t^4\) is a power function.

The correct option is the second one.

Answer:

• (A) \(y = 1200(0.96)^t\)
• (B) \(y = 1200(1.04)^t\) (Correct answer)
• (C) \(y = 1200 + 0.04t\)
• (D) \(y = 1200t^4\)