Question

Write an exponential decay model for each situation. The value of x for each value of f(x) will lie between two consecutive whole numbers. List the whole numbers.

initial value: 1,800
decay rate: 7%
f(x) = 400

Answer 1

An exponential decay model can be formulated using the given initial value and decay rate, and the time when a specific reduced value is reached can be calculated and will lie between two whole numbers.

Step-by-step explanation:

To model exponential decay, we use the formula f(x) = a * e(-kx), where a is the initial value, k is the decay rate, and x represents time. For the given situation, the initial value a is 1,800, and the decay rate k is 7% or 0.07. To find the model when f(x) is 400, we set up the equation 400 = 1800 * e(-0.07x).

To solve for x, we take the natural logarithm of both sides, resulting in ln(400/1800) = -0.07x. Simplifying further, we divide by -0.07 to isolate x. After calculating, we find that x falls between two whole numbers, which can then be determined.

Answer 2

The numbers are;

20 and 21

Step-by-step explanation:

A general form of an exponential decay model can be presented as follows;

f(x) = a·(1 + r)ˣ

Where;

f(x) = The output for a given input variable, 'x'

x = The input variable

a = The initial value

r = The growth (+ve) or decay (-ve) rate

When f(x) = 400, a = 1,800, r = -7% = -0.07, we get;

400 = 1,800 × (1 - 0.07)ˣ

(1 - 0.07)ˣ = 400/1,800 = 2/9

ln((1 - 0.07)ˣ) = ln(2/9)

x = ln(2/9)/(ln(1 - 0.07)) ≈ 20.73

The value of 'x' lies between 20 and 21.