Suppose you're going to sell a product, and your marketing team has determined that the maximum number of units of the product that can be sold is given by a constant M>0 and that the rate of increase in unit sales will be proportional to the difference between M and the number of units that have currently been sold. Then the cumulative number of units M>0 sold for any time t will be given by a function of the form:
A. U(t) = Ae^(-kt) - M
A. U(t) = Ae^(kt)
A. U(t) = Ae^(-kt) + M
A. U(t) = Ae^(kt) - M
A. U(t) = Ae^(kt) + M
(A is some undetermined constant, and k>0 is some proportionality constant)
you know that
du/dt = k*(M-u)
du/(M-u) = k dt
du/(u-M) = -k dt
ln(u-M) = -kt + c
u-M = e^c * e^(-kt)
u = Ae^(-kt) + M
Oops the answer choices should be A - E
A. U(t) = Ae^(-kt) + M
The correct option is: A. U(t) = Ae^(-kt) + M
Explanation:
Based on the information given, the rate of increase in unit sales is proportional to the difference between M and the number of units that have currently been sold. This implies that the rate of increase in unit sales can be expressed as -k(U(t) - M), where k is the proportionality constant.
To find the cumulative number of units sold at any time t, we need to integrate the rate of increase function over time. Therefore, the equation becomes:
dU/dt = -k(U - M)
Rearranging the equation and separating variables, we get:
-(1/(U - M)) dU = k dt
Integrating both sides, we have:
-∫(1/(U - M)) dU = ∫k dt
This simplifies to:
-ln(U - M) = kt + C
where C is the constant of integration.
To find the value of C, we can plug in the initial condition that U(0) = 0 (assuming no units have been sold at the starting point). This yields:
-ln(0 - M) = k(0) + C
-ln(-M) = C
Therefore, the equation becomes:
-ln(U - M) = kt - ln(-M)
Rearranging further, we get:
ln(U - M) = -kt + ln(-M)
Exponentiating both sides, we have:
U - M = e^(-kt) * -M
Finally, rearranging the equation, we get:
U = Ae^(-kt) + M
where A = -M is an undetermined constant.
Hence, the correct form of the equation for the cumulative number of units sold is U(t) = Ae^(-kt) + M.
To determine the correct function that represents the cumulative number of units sold over time, we need to analyze and interpret the given information.
1. The maximum number of units that can be sold is given by a constant M > 0. This means that the cumulative number of units sold will approach M as time goes to infinity.
2. The rate of increase in unit sales is proportional to the difference between M and the current number of units sold. This implies that as the number of units sold gets closer to M, the rate of increase in sales decreases.
Based on this information, we can derive the correct function representing the cumulative number of units sold over time:
Let's represent the cumulative number of units sold at time t as U(t). According to the information provided, U(t) should be proportional to the difference between M and the number of units sold at time t. The rate of proportionality is given by some constant k > 0.
Therefore, we can write the equation as:
U(t) = A * e^(-kt) + M
Where:
- U(t) represents the cumulative number of units sold at time t.
- A is an undetermined constant, which represents the initial number of units sold.
- e is the base of the natural logarithm, approximately 2.71828.
- k is a proportionality constant, which determines the rate of increase.
Hence, the correct answer is:
A. U(t) = Ae^(-kt) + M