The value, V, of Kalani’s stock investments over a time period, x, can be determined using the equation V=75-(0.80)^-x What is the rate of increase or decrease associated with this account?
25% DECREASE
To determine the rate of increase or decrease associated with Kalani's stock investments, we need to calculate the derivative of the equation V = 75 - (0.80)^(-x) with respect to time, x.
Let's start by expressing the equation with a positive exponent instead of a negative one:
V = 75 - (1 / (0.80)^x)
To find the derivative, we apply the power rule. The derivative of a constant, such as 75, is zero. The derivative of (1 / (0.80)^x) is calculated as follows:
dV/dx = -ln(0.80) * (1 / (0.80)^x)
Simplifying this expression gives:
dV/dx = -ln(0.80) / (0.80)^x
So, the rate of increase or decrease associated with this account is given by the derivative expression:
Rate = -ln(0.80) / (0.80)^x
To determine the rate of increase or decrease associated with this account, we need to analyze the equation for V in terms of x.
The equation for V is given as V = 75 - (0.80)^(-x).
At first glance, it might seem unclear how to determine the rate of increase or decrease directly from this equation. However, we can identify some key pieces of information that will help us understand how V changes with respect to x.
1. The term (0.80)^(-x) represents a number raised to a negative power. This means that as x increases, the value of (0.80)^(-x) will decrease.
2. 75 is a constant value in the equation and does not change with x.
Given these details, we can conclude the following:
- As x increases, the term (0.80)^(-x) decreases, since raising a number to a negative power makes it smaller.
- The decrease in (0.80)^(-x) causes V to increase because it is subtracted from the constant value 75 in the equation.
Therefore, we can say that as x increases, the value of V will increase. This indicates a positive rate of increase associated with the account.
To summarize, the rate of increase associated with this account is positive, as the value of V increases when x increases.