Universal instruments found that the monthly demand for its new line of Galaxy Home Computers t months after placing the line on the market was given by
D(t) = 2900 − 2300e−0.08t (t > 0)
Graph this function and answer the following questions.
(a) What is the demand after 1 month? After 1 year? After 2 years? after 5 years?
after 1 month computers
after 1 year computers
after 2 years computers
after 5 years computers
(b) At what level is the demand expected to stabilize?
computers
(c) Find the rate of growth of the demand after the tenth month.
computers per month
hmmmm confusing
fcdsc
To graph the function D(t) = 2900 - 2300e^(-0.08t), we can follow these steps:
1. Choose values for t: In order to plot the graph, we need to select specific values for t. Let's choose t values that are evenly spaced, such as 0, 1, 2, 3, and so on.
2. Calculate D(t): Plug the chosen values of t into the function D(t) = 2900 - 2300e^(-0.08t) to find the corresponding values of D(t). For example, if t = 0, then D(0) = 2900 - 2300e^(-0.08 * 0) = 2900 - 2300e^0 = 2900 - 2300(1) = 2900 - 2300 = 600.
3. Plot the points: Plot the points (t, D(t)) on a graph. Repeat this for all the chosen values of t.
4. Connect the points: Draw a smooth curve through the plotted points to create the graph of the function.
Now, let's find the demand for different values of t:
(a) After 1 month:
Plug t = 1 into the function: D(1) = 2900 - 2300e^(-0.08 * 1)
Evaluate: D(1) = 2900 - 2300e^(-0.08) ≈ 2900 - 2300(0.923) ≈ 2900 - 2122 = 778 computers
After 1 year:
Plug t = 12 into the function: D(12) = 2900 - 2300e^(-0.08 * 12)
Evaluate: D(12) = 2900 - 2300e^(-0.96) ≈ 2900 - 2300(0.383) ≈ 2900 - 881 = 2019 computers
After 2 years:
Plug t = 24 into the function: D(24) = 2900 - 2300e^(-0.08 * 24)
Evaluate: D(24) = 2900 - 2300e^(-1.92) ≈ 2900 - 2300(0.146) ≈ 2900 - 335 = 2565 computers
After 5 years:
Plug t = 60 into the function: D(60) = 2900 - 2300e^(-0.08 * 60)
Evaluate: D(60) = 2900 - 2300e^(-4.8) ≈ 2900 - 2300(0.006) ≈ 2900 - 14 = 2886 computers
(b) To find the level at which the demand is expected to stabilize, we need to find the value of D(t) as t approaches infinity. As t approaches infinity, the term e^(-0.08t) becomes negligible, and the demand converges to a constant value.
As t approaches infinity: D(t) = 2900 - 2300e^(-0.08t) ≈ 2900 - 0 ≈ 2900 computers
So, the demand is expected to stabilize at 2900 computers.
(c) The rate of growth of the demand after the tenth month can be calculated by finding the derivative of the demand function D(t) with respect to t. The derivative gives us the rate of change or the slope of the function at a specific point.
Differentiating D(t) = 2900 - 2300e^(-0.08t) with respect to t:
D'(t) = 0 - (-2300)(-0.08)e^(-0.08t) ≈ 184e^(-0.08t)
After the tenth month, let t = 10:
D'(10) ≈ 184e^(-0.08 * 10) = 184e^(-0.8) ≈ 184(0.449) ≈ 82.73 computers per month
Therefore, the rate of growth of demand after the tenth month is approximately 82.73 computers per month.