/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 70 (a) use the position equation \(... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 70

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2},\) (d) describe the slope of the secant line through \(t_{1}\) and \(t_{2},\) (e) find the equation of the secant line through \(t_{1}\) and \(t_{2},\) and (f) graph the secant line in the same viewing window as your position function. An object is dropped from a height of 80 feet. \(t_{1}=1, t_{2}=2\)

Short Answer

Expert verified
Position function: \(s(t)=-16t^{2}+80\). The graph of this function is a downward parabola starting at (0,80). Average rate of change from \(t_{1}\) to \(t_{2}\) is the slope of the secant line which represents the average velocity of the object in that interval. The equation of the secant line can be obtained using the point-slope formula and plotted in the same view as the position function.

Step by step solution

01

Formulate Position Function

Substitute \(s_{0}\) (initial height) = 80, \(v_{0}\) (initial velocity) = 0 (since the object is dropped and not thrown) in the given position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to get the position function \(s(t)=-16t^{2} +80\).
02

Graph the Position Function

By using a graphing utility, plot \(s(t)=-16t^{2}+80\). The graph is a downward parabola starting at (0,80).
03

Compute the Average Rate of Change

The average rate of change from \(t_{1}\) to \(t_{2}\) is calculated using the formula \(\frac{\Delta s}{\Delta t}\) where \(\Delta s = s(t_{2})- s(t_{1})\) and \(\Delta t = t_{2}-t_{1}\). Substitute \(t_{1}=1, t_{2}=2\) in our position function to obtain \(s(1)\) and \(s(2)\), and use these to calculate the rate of change.
04

Interpret the Slope

The slope of the secant line through \(t_{1}\) and \(t_{2}\) is same as the average rate of change, it represents the average velocity of the object during the time interval from \(t_{1}\) to \(t_{2}\).
05

Find the Equation of Secant Line

Using the point-slope formula \(y - y_{1} = m(x - x_{1})\), where m is the slope and (x_{1}, y_{1}) is a point on the line, we can find the equation of the secant line. Use \(m = \frac{\Delta s}{\Delta t}\) for slope, and pick either (\(t_{1}, s(t_{1})\)) or (\(t_{2},s(t_{2})\)) as the point.
06

Graph the Secant Line

Graph the equation derived in step 5 using the same graphing utility. The secant line will cut through the parabola at \(t_{1}\) and \(t_{2}\).

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