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Let /(x) = 1 + 2x – 22. Evaluate the following and simplify as much as possible. (a) f(((((O))))) f (x + h) – $(x) (b) S (3. - 1) h (c) 7. A window is to be made in the shape of a rectangle with a semi-circle on top, so that the diameter of the semi-circle equals the width of the rectangle. If the perimeter of the window is 40 feet and the width of the rectangle is a feet, find a formula for Ax) the area of the window in square feet. 8. The population of New York state was 19 million in 2000, and 23 million in 2020.

(a) What was the average rate of change in the population between 2000 and 2020? (b) Assuming a linear growth model for the population, find a formula for PCt), the population of New York t years after 2000. (c) Use your formula for PC) to predict the population in 2045.

Answer :

The formula for the area of the described window shape is a*(20-a/2) + π*(a/2)²/2. The average population change in New York between 2000 and 2020 was 0.2 million people per year, and the forecasted population for 2045, assuming linear growth, is 28 million people.

Firstly, it's not clear what the functions /(x), $(x), and S(x) represent due to possible typos, so I won't attempt to evaluate those parts of the question. Instead, I'll focus on the areas of the question which are clear.

In the problem with the window, if the width of the rectangle (which is also the diameter of semi-circle) is a feet and perimeter of the window is 40 feet, the formula for the area (Ax) of the window would be a*(20-a/2) + π*(a/2)²/2. The first term represents the area of the rectangle while the latter represents the area of a semi-circle.

For the population change in New York, the average rate of change = (23–19) million people/20 years = 0.2 million people per year. If we assume linear growth, we get PC(t) = 19 + 0.2t, where t is years after 2000. Using this formula, the estimated population in 2045 will be PC(45) = 19 + 0.2*45 = 28 million people.

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