The number of banks in a country for the years 1935 through 2009 is given by the following function.
f(x)= 83.5x+12,368 if x<90 −376.5x+48,687 if x≥90 , where x is the number of years after 1900
Complete parts (a)-(b).
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Part 1
a) What does this model give as the number of banks in 1970? 2000? 2030?

To find the number of banks in a particular year, substitute the corresponding value of $x$ into the given function $f(x)$.

a) For 1970, $x = 1970 - 1900 = 70$:
\[ f(70) = 83.5(70) + 12,368 \]

b) For 2000, $x = 2000 - 1900 = 100$:
\[ f(100) = -376.5(100) + 48,687 \]

c) For 2030, $x = 2030 - 1900 = 130$:
\[ f(130) = -376.5(130) + 48,687 \]

Calculate these values to find the respective number of banks for the given years.

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sqdancefan sqdancefan · Answer 2 · 2023-12-23T00:26:08+01:00

Answer:

1970: 18213
2000: 11037
2030: undefined

Step-by-step explanation:

Given the piecewise function for the number of banks from 1935 to 2009, you want to know what the model predicts for the number of banks in 1970, 2000, and 2030.

[tex]f(x)=\begin{cases}83.5x+12368&\text{for }35\le x < 90\\-376.5x+48687&\text{for }90\le x\le109\end{cases}[/tex]

1970

For 1970, the value of x is 1970 -1900 = 70. The model predicts the number of banks is ...

f(70) = 83.5(70) +12368 = 18213

The number of banks in 1970 is given as 18213.

2000

For 2000, the value of x is 2000 -1900 = 100. The model predicts the number of banks is ...

f(100) = -376.5(100) +48687 = 11037

The number of banks in 2000 is given as 11037.

2030

For 2030, the value of x is 2030 -1900 = 130. We note that the model is only defined for years up to 2009, so it does not give a prediction for the year 2030.

The number of banks given by the model for the year 2030 is undefined.

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Additional comment

If you evaluate the model using x = 130 and the second piece of the definition, you get the value -258. The number of banks is zero in the year 2029, and goes negative after that, according to the model.

We note, too, there is a significant discontinuity at x=90, suggesting sudden failure of of about 5081 banks at the end of 1989.