Annual sales (in millions of units) of pocket computers are expected to grow in accordance with the following function where t is measured in years, with t = 0 corresponding to 1997.
f(t) = 0.18t^2 + 0.16t + 2.64
How many pocket computers will be sold over the 2 year period between the beginning of 1997 and the end of 1998?
NOTE: I got 1.04 but it says it is wrong.
Hmmm. we have
∫[0,2] 0.18t^2 + 0.16t + 2.64
= 0.06t^3 + 0.08t^2 + 2.64t [0,2]
= .48 + .32 + 5.28
= 6.08
What did you do to arrive at your answer?
1.04 is obviously way off, since f(0) = 2.64 just to start with.
I did my math wrong, I don't know what I was thinking. Thanks for your help.
To find the number of pocket computers sold over the 2-year period between the beginning of 1997 and the end of 1998, we need to calculate the value of the integral of the function f(t) over that period.
First, we need to determine the limits of integration. In this case, the beginning of 1997 corresponds to t = 0, and the end of 1998 corresponds to t = 2.
The integral of the function f(t) over the interval [0, 2] is given by:
∫[0 to 2] (0.18t^2 + 0.16t + 2.64) dt
To simplify the calculation, we can find the antiderivative of each term separately and evaluate the integral.
The antiderivative of 0.18t^2 is (0.18/3)t^3 = 0.06t^3.
The antiderivative of 0.16t is (0.16/2)t^2 = 0.08t^2.
The antiderivative of 2.64 is 2.64t.
Applying the antiderivatives and evaluating the integral over the interval [0, 2], we have:
[0.06t^3 + 0.08t^2 + 2.64t] evaluated from 0 to 2
Plugging in the upper limit (2) into the expression and subtracting the result when the lower limit (0) is plugged in, we get:
[0.06(2)^3 + 0.08(2)^2 + 2.64(2)] - [0.06(0)^3 + 0.08(0)^2 + 2.64(0)]
= [0.06(8) + 0.08(4) + 2.64(2)] - [0 + 0 + 0]
= [0.48 + 0.32 + 5.28] - [0 + 0 + 0]
= 6.08 - 0
= 6.08
Therefore, the number of pocket computers that will be sold over the 2-year period between the beginning of 1997 and the end of 1998 is 6.08 million units.
To find the number of pocket computers that will be sold over the 2-year period between the beginning of 1997 and the end of 1998, we need to calculate the difference in sales between those two years.
First, we need to determine the value of t for the beginning of 1997 and the end of 1998:
For the beginning of 1997, t = 0 (since t = 0 corresponds to 1997).
For the end of 1998, t = 1998 - 1997 = 1.
Now, we can substitute these values of t into the given function f(t) = 0.18t^2 + 0.16t + 2.64:
For the beginning of 1997 (t = 0):
f(0) = 0.18(0)^2 + 0.16(0) + 2.64
= 0 + 0 + 2.64
= 2.64 million units
For the end of 1998 (t = 1):
f(1) = 0.18(1)^2 + 0.16(1) + 2.64
= 0.18 + 0.16 + 2.64
= 2.98 million units
To find the number of pocket computers sold over the 2-year period, we need to calculate the difference between the sales at the end of 1998 and the beginning of 1997:
Number of pocket computers sold = f(1) - f(0)
= 2.98 - 2.64
= 0.34 million units.
Therefore, the correct answer is 0.34 million units, not 1.04.