Question

The enrollment growth of a trade school has exact inverse variation with the economic growth of the region. If the enrollment of the trade school now is 1200, what is the projected student enrollment in the school in the year 2030?

Answer 1

The projected student enrollment in the trade school in the year 2030 is 800.

Step-by-step explanation:

The relationship between the enrollment growth of the trade school and the economic growth of the region is described as exact inverse variation. In mathematical terms, this can be represented as
`\(y = (k)/(x)\),` where
`\(y\)` is the enrollment,
`\(x\)` is the economic growth, and
`\(k\)` is a constant of variation.

Given that the current enrollment
`(\(y\))` is 1200, we can set up the equation as
`\(1200 = (k)/(x)\).` To find
`\(k\),` we need additional information about the economic growth. However, since we are interested in the projected enrollment in 2030, we can use the inverse variation relationship to make an inference.

If the economic growth is expected to increase, the enrollment will decrease, and vice versa. In this case, if we assume a moderate increase in economic growth by 50% (1.5 times the current value), we can calculate the projected enrollment for 2030. Let
`\(x_(2030)\)` represent the economic growth in 2030.

`\[ 1200 = (k)/(1.5x_(2030)) \]`

Solving for
`\(x_(2030)\),` we find that
`\(x_(2030) = (k)/(1.5 * 1200)\).` Assuming
`\(k\)` remains constant, we can calculate the projected enrollment by substituting
`\(x_(2030)\)` into the inverse variation equation.

`\[ y_(2030) = (k)/(x_(2030)) \]`

Substituting the values, we get
`\(y_(2030) = (k)/((k)/(1.5 * 1200)) = 800\).` Therefore, the projected student enrollment in the trade school in the year 2030 is 800, considering the exact inverse variation between enrollment and economic growth.