A contractor is calculating the number of decorative stones for a new rectangular patio. The bids always include a count for the number of stones for the edging, E(x), and the number of stones for the center of the patio, C(x). The various patio sizes have lengths represented by the function f(x) and widths represented by g(x).

Which statement describes the combined functions E(x) and C(x)?

The number of edge stones needed has a constant rate of change, but the number of stones for the center does not.
The number of stones for the center has a constant rate of change but the number of edge stones needed does not.
Both the number of stones needed for the edge and the number of stones for the center have a constant rate of change.
Neither the number of stones needed for the edge nor the number of stones for the center has a constant rate of change

Two functions E(x) and C(x) are given in the table along with some of the function values. We have to identify if any of these functions show constant rate of change or not.

By constant rate of change we mean that the slope is constant or in other words, a linear relationship is shown by the function. For a Linear function, the first differences of the function values are same.

By first differences we mean the difference between two consecutive output values. We can see that difference between consecutive input values is constant i.e. 1. If the difference between consecutive output values of any of the functions is same, then that function will be a Linear Function and, therefore, the rate of change for that function will be constant.

Lets analyze E(x) first. The output values are:

34, 52, 70 and 88

The difference between consecutive output values is:

18, 18 and 18

Since, this difference is constant, we can conclude that E(x) is a Linear function with a constant rate of change.

Now lets analyze C(x). The output values are:

62, 133, 260 and 435

The difference between consecutive output values is:

71, 127 and 175

Since, the differences are not constant, C(x) is not a Linear Function and , therefore, the rate of change of C(x) is not constant.

Conclusion:

The number of edge stones which is represented by E(x) has a constant rate of change, but the number of stones for center which is represented by C(x) does not have constant rate of change. This makes the first option our correct answer.