Questions: Consider the position function (s(t)=sin(pi t)) representing the position of an object moving along a line on the end of a spring. Sketch a graph of (s) together with the secant line passing through ((0, s(0))) and ((0.9, s(0.9))). Determine the slope of the secant line and explain its relationship to the moving object. Determine the slope of the secant line. (msec=0.343) (Round to the nearest thousandth as needed.) What is the relationship of the slope to the velocity of the moving object? A. The slope is the average velocity of the object over the interval ([0,0.9]). B. The slope is the maximum velocity of the object. C. The slope is the instantaneous velocity when (t=0.9). D. The slope is the average velocity of the object over the interval ([0,0.309]).

Consider the position function (s(t)=sin(pi t)) representing the position of an object moving along a line on the end of a spring. Sketch a graph of (s) together with the secant line passing through ((0, s(0))) and ((0.9, s(0.9))). Determine the slope of the secant line and explain its relationship to the moving object. Determine the slope of the secant line. (msec=0.343) (Round to the nearest thousandth as needed.) What is the relationship of the slope to the velocity of the moving object? A. The slope is the average velocity of the object over the interval ([0,0.9]). B. The slope is the maximum velocity of the object. C. The slope is the instantaneous velocity when (t=0.9). D. The slope is the average velocity of the object over the interval ([0,0.309]).
Transcript text: Consider the position function $\mathrm{s}(\mathrm{t})=\boldsymbol{\operatorname { s i n }}(\boldsymbol{\pi} \mathrm{t})$ representing the position of an object moving along a line on the end of a spring. Sketch a graph of $s$ together with the secant line passing through $(0, s(0))$ and $(0.9, s(0.9))$. Determine the slope of the secant line and explain its relationship to the moving object. Determine the slope of the secant line. $m_{s e c}=0.343$ (Round to the nearest thousandth as needed.) What is the relationship of the slope to the velocity of the moving object? A. The slope is the average velocity of the object over the interval $[0,0.9]$. B. The slope is the maximum velocity of the object. C. The slope is the instantaneous velocity when $t=0.9$. D. The slope is the average velocity of the object over the interval $[0,0.309]$.
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Solution

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Solution Steps

Step 1: Identify the correct graph

The graph of \(s(t) = \sin(\pi t)\) passes through (0,0) and (0.9, sin(0.9π)). Since sin(0.9π) is positive, the second point will be in the first quadrant. The correct graph showing the secant line is D.

Step 2: Calculate the slope of the secant line

The secant line passes through the points (0, s(0)) and (0.9, s(0.9)). s(0) = sin(π * 0) = sin(0) = 0 s(0.9) = sin(π * 0.9) = sin(0.9π) The slope, m_sec, is given by: \(m_{sec} = \frac{s(0.9) - s(0)}{0.9 - 0} = \frac{\sin(0.9\pi) - 0}{0.9} = \frac{\sin(0.9\pi)}{0.9} \approx \frac{0.309}{0.9} \approx 0.343\)

Step 3: Interpret the meaning of the slope

The slope of the secant line represents the average velocity of the object over the interval [0, 0.9].

Final Answer

The correct graph is D. The slope of the secant line is approximately \\(\boxed{0.343}\\). This slope represents the average velocity of the object over the interval [0, 0.9].

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