The rate of spending on grants by foundations in the period of 1993-2003 was approximately s(t)=11+20/(1+1,800e^(-0.9t) billion dollars per year with t being the number of years since 1990 and (t being greater than or equal to 3 while also being less than of equal to 13). How can I estimate the total spending to the nearest $10 billion dollars on grants from 1998 to 2003?
since s(t) is the rate of spending, the amount spent is just
∫ s(t) dt
over the interval from t = 5 to 10
So I would just find ∫ [5,10] 11+20/(1+1,800e^(-0.9t)?
Why wouldn't the interval limits be the 3 and 13?
because you didn't read the question carefully.
To estimate the total spending on grants from 1998 to 2003, we need to calculate the integral of the rate function over the given time interval. Since the rate function is given as s(t) = 11 + 20/(1 + 1,800e^(-0.9t)), we will be integrating s(t) with respect to t from t = 8 to t = 13.
To calculate the integral, we can use the definite integral formula. Let's break down the steps:
1. Start by writing the integral:
∫[8 to 13] (11 + 20/(1 + 1,800e^(-0.9t))) dt
2. Next, simplify the integral:
∫[8 to 13] 11 + 20/(1 + 1,800e^(-0.9t)) dt
3. To evaluate this integral, we can attempt a substitution. Let's substitute u = 1 + 1,800e^(-0.9t). Then, du = (-0.9)(1,800)e^(-0.9t) dt, which simplifies to du = -1,620e^(-0.9t) dt.
4. Now, substitute the variables:
∫[8 to 13] 11 + 20/u * du
5. Integrate this expression:
[11t + 20ln|u|] [8 to 13]
6. Plug in the values of the limits of integration (13 and 8):
(11 * 13 + 20ln|u|(13)) - (11 * 8 + 20ln|u|(8))
7. Substitute back u = 1 + 1,800e^(-0.9t):
(11 * 13 + 20ln|1 + 1,800e^(-0.9t)|(13)) - (11 * 8 + 20ln|1 + 1,800e^(-0.9t)|(8))
8. Once you substitute the values and simplify, you will get the estimated total spending on grants from 1998 to 2003.
Remember to round the result to the nearest $10 billion dollars as requested in the question.