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A corporation uses a model of diminishing returns to make predictions about the expected returns on research investment. For this model, in order to produce an x % increase in annual profits in subsequent years, the corporation must invest y % of annual profits into research, where \(\mathrm{y} = 2\mathrm{x}^2\).
Select two different numbers that are jointly compatible with the information provided and could be the values for \(\mathrm{x}\) and for \(\mathrm{y}\). Make only two selections, one in each column.
1
3
5
20
50
80
We have a diminishing returns model where:
Let's create a simple table to organize our thinking:
| Variable | Represents | Formula Role |
| x | % profit increase | Input |
| y | % invested in research | Output = \(\mathrm{2x^2}\) |
If a company wants a 10% profit increase (\(\mathrm{x = 10}\)), they must invest:
\(\mathrm{y = 2(10)^2 = 2(100) = 200\%}\) of annual profits
This shows the "diminishing returns" - higher profit goals require disproportionately higher research investment.
We need to find:
These values must:
We're looking for a pair (x, y) where plugging x into our formula gives us a y value that's also in our answer choices.
Let's test each possible x value to see if it produces a y value in our choices:
If \(\mathrm{x = 1}\):
\(\mathrm{y = 2(1)^2 = 2(1) = 2}\)
Is 2 in our choices? No, continue.
If \(\mathrm{x = 3}\):
\(\mathrm{y = 2(3)^2 = 2(9) = 18}\)
Is 18 in our choices? No, continue.
If \(\mathrm{x = 5}\):
\(\mathrm{y = 2(5)^2 = 2(25) = 50}\)
Is 50 in our choices? Yes! ✓
Stop here - we found our answer.
With x = 5% profit increase, the company must invest y = 50% of annual profits into research. This satisfies our equation \(\mathrm{y = 2x^2}\) and both values are in our answer choices.