Monthly sales, in thousands, of a new laptop computer grow according to the model
N(t)=4.7te^-0.3t
where t is the number after introduction. Determine the point of diminishing returns and interpret your answer.
well, what does diminishing returns mean? Sales decrease while numbers increase.
So, you want to find t for max N.
the point at which the growth of a function begins to slow down
good call. So, rather than being where f' goes negative, it is where f' starts decreasing: where f" goes negative. That is, when the graph changes from concave up to concave down.
Unfortunately, this graph starts out concave down, so the growth rate is always decreasing, and then it starts increasing at t = 20/3
http://www.wolframalpha.com/input/?i=4.7te%5E(-0.3t)+,+0%3Ct%3C20
ty!
To determine the point of diminishing returns for the monthly sales of the laptop computer, we need to find the point at which the rate of increase starts to slow down significantly.
The rate of increase can be determined by taking the derivative of the monthly sales function N(t) with respect to t:
N'(t) = (4.7)e^(-0.3t) - 0.3(4.7)te^(-0.3t)
To find the point of diminishing returns, we set the derivative equal to zero and solve for t:
(4.7)e^(-0.3t) - 0.3(4.7)te^(-0.3t) = 0
Dividing by (4.7)e^(-0.3t), we get:
1 - 0.3t = 0
Simplifying further, we have:
0.3t = 1
t = 1 / 0.3
t ≈ 3.33
Therefore, the point of diminishing returns occurs approximately 3.33 units of time after the introduction of the laptop computer.
Interpretation:
At the point of diminishing returns, the rate of increase in monthly sales starts to slow down significantly. This means that after around 3.33 units of time, the growth of monthly sales becomes less significant or less rapid. This could imply that the market is becoming saturated as more people already have the laptop or there is increased competition from other similar products. The company may need to implement new marketing strategies or introduce updated features to maintain and further improve sales.