Question

number of hours (x) | cost in dollars (y)
1 | 65
2 | 105
3 | 145
4 | 185
the relationship between the number of hours a national park guide is hired and the cost in dollars for the national park guide’s services is shown in the table.
which statement is true?
b. the rate of change is 40.
a. the function is nonlinear.
d. the rate of change is 25.
c. the y-intercept is 65.

Explanation:

Step1: Check if function is linear

Calculate the rate of change (slope) between consecutive points. For \(x = 1\) to \(x = 2\): \(\frac{105 - 65}{2 - 1}=\frac{40}{1}=40\). For \(x = 2\) to \(x = 3\): \(\frac{145 - 105}{3 - 2}=\frac{40}{1}=40\). For \(x = 3\) to \(x = 4\): \(\frac{185 - 145}{4 - 3}=\frac{40}{1}=40\). Since the rate of change is constant, the function is linear. So option A is false.

Step2: Determine the rate of change

From Step 1, the rate of change (slope) is 40. So option B says "The rate of change is 40" which matches. Option D says 25, which is incorrect.

Step3: Find the y - intercept

The linear equation is \(y=mx + b\), where \(m = 40\). Using the point \((1,65)\): \(65=40(1)+b\). Solving for \(b\): \(b=65 - 40=25\). So the y - intercept is 25, not 65. So option C is false.

Answer:

B. The rate of change is 40.