Question

consider the position function ( s(t) = -16t^2 + 112t ) representing the position of an object moving vertically along a line. determine the slope of the secant line (passing through ( (0, s(0)) ) and ( (2, s(2)) )) and explain the relationship to the moving object. match the graph of ( s ) with the secant line. choose the correct graph below (options a, b, c with graphs).

Explanation:

Step1: Recall the secant line formula

The slope of the secant line between two points \((t_1, s(t_1))\) and \((t_2, s(t_2))\) is given by \(m=\frac{s(t_2)-s(t_1)}{t_2 - t_1}\). Here, the function is \(s(t)=- 16t^{2}+121\), and we consider the points \(t = 0\) and \(t = 2\) (assuming from the graph context, since the time axis has \(t\) and the vertical axis is \(s\)).

First, find \(s(0)\) and \(s(2)\):

• For \(t = 0\): \(s(0)=-16(0)^{2}+121 = 121\)
• For \(t=2\): \(s(2)=-16(2)^{2}+121=-16\times4 + 121=-64 + 121 = 57\)

Step2: Calculate the slope of the secant line

Using the slope formula \(m=\frac{s(2)-s(0)}{2 - 0}=\frac{57 - 121}{2}=\frac{- 64}{2}=-32\)

Now, we analyze the graphs. The secant line connects \((0,121)\) and \((2,57)\). We need to check which graph has the secant line with this slope. The slope is negative, and the line should go from \((0,121)\) (top) to \((2,57)\) (lower). By comparing the graphs, we can see that the correct graph should have the secant line with this slope.

Answer:

(Assuming the correct graph is, for example, Graph A (depending on the actual graph's secant line slope and points, but following the calculation, the slope is -32, so the graph where the secant line connects (0,121) and (2,57) with slope -32 is the correct one. If we assume the options are A, B, C and the correct one is A, then) A