The rate at which iPads are sold is R(t)=1000000x2.8^t where the units of R(t) are millions/year and represents time in years such that the year 2010 (Jan 1 2010 through Dec 31 2010) corresponds to t=0. How many iPads will be sold in the three year period Jan 1, 2012 through Dec 31, 2014?
To find out how many iPads will be sold in the three-year period from Jan 1, 2012, through Dec 31, 2014, we need to calculate the total number of iPads sold during this period.
The given function represents the rate at which iPads are sold over time. R(t) = 1000000 * 2.8^t, where t is the time in years.
To calculate the total number of iPads sold during a specific time period, we need to integrate the rate function over that period. The integral of the rate function will give us the total number of iPads sold.
Since we are interested in the three-year period from Jan 1, 2012, through Dec 31, 2014, we need to find the integral of the rate function R(t) over this interval.
Let's break down the calculation step-by-step:
Step 1: Convert the given period into a suitable range of t-values:
We know that t = 0 corresponds to the year 2010. So, to calculate the number of iPads sold in the three-year period from Jan 1, 2012, through Dec 31, 2014, we need to find the corresponding t-values for these dates.
Since the year 2012 is two years after 2010, the t-value for Jan 1, 2012, will be t = 2.
Similarly, the year 2014 is four years after 2010, so the t-value for Dec 31, 2014, will be t = 4.
Therefore, we need to integrate the rate function R(t) over the interval t = 2 to t = 4.
Step 2: Set up the integral:
The integral of the rate function R(t) over the interval [a, b] represents the total number of iPads sold from time t = a to t = b.
In our case, the interval is from t = 2 to t = 4. So, the integral we need to evaluate is:
∫[2,4] R(t) dt
Step 3: Evaluate the integral:
To evaluate the integral, we can use the power rule of integration.
Using the power rule, the integral of 2.8^t with respect to t is (1/ln(2.8)) * 2.8^t.
Applying the power rule to the entire integrand, we get:
∫[2,4] R(t) dt = ∫[2,4] 1000000 * 2.8^t dt = 1000000 * (1/ln(2.8)) * [2.8^t] evaluated from t = 2 to t = 4
Step 4: Calculate the integral:
Evaluate the integral by plugging in the upper and lower limits of integration:
∫[2,4] R(t) dt = 1000000 * (1/ln(2.8)) * [2.8^4 - 2.8^2]
Now, we can substitute the values into the equation:
∫[2,4] R(t) dt ≈ 1000000 * (1/ln(2.8)) * [2.8^4 - 2.8^2] ≈ 1000000 * (1/ln(2.8)) * [97.7344 - 8.40704]
Calculating the expression within the brackets:
[97.7344 - 8.40704] ≈ 89.32736
Step 5: Calculate the total number of iPads sold:
Substituting the calculated value back into the equation:
∫[2,4] R(t) dt ≈ 1000000 * (1/ln(2.8)) * 89.32736 ≈ 32176633.77
So, approximately 32,176,634 iPads will be sold in the three-year period from Jan 1, 2012, through Dec 31, 2014.