Chapter 1.4, Problem 7E

a.

To determine

To find: The slope of the secants $P{Q}_1,P{Q}_2,P{Q}_3$ and $P{Q}_4$ .

Answer to Problem 7E

  $\begin{array}{cc}\text{Secant}& \text{Slope}\\ P{Q}_1& 43\\ P{Q}_2& 46\\ P{Q}_3& 50\\ P{Q}_4& 50\end{array}$

Units for the slopes will be meters/second.

Explanation of Solution

Given information:

  

The points are: $Q_1=\left(10,225\right),Q_2=\left(14,375\right),Q_3=\left(16.5,475\right),Q_4\left(18,550\right)\text{ and }P=\left(20,650\right)$

Concept used:

The slope of secant between the points $\left(a,f\left(a\right)\right)\text{ and }\left(b,f\left(b\right)\right)$ is given by: $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

Calculation:

Slope of the secant $P{Q}_1\text{ where }{Q}_1=\left(10,225\right)\text{ and }P=\left(20,650\right)$ is:

  $\begin{array}{c}P{Q}_1=\frac{650-225}{20-10}\\ =\frac{425}{10}\\ =42.5\\ \approx 43\end{array}$

Slope of the secant $P{Q}_2\text{ where }{Q}_2=\left(14,357\right)\text{ and }P=\left(20,650\right)$ is:

  $\begin{array}{c}P{Q}_2=\frac{650-375}{20-14}\\ =\frac{275}{6}\\ =45.833\\ \approx 46\end{array}$

Slope of the secant $P{Q}_3\text{ where }{Q}_3=\left(16.5,475\right)\text{ and }P=\left(20,650\right)$ is:

  $\begin{array}{c}P{Q}_3=\frac{650-475}{20-16.5}\\ =\frac{175}{3.5}\\ =50\end{array}$

Slope of the secant $P{Q}_4\text{ where }{Q}_4=\left(18,550\right)\text{ and }P=\left(20,650\right)$ is:

  $\begin{array}{c}P{Q}_4=\frac{650-550}{20-18}\\ =\frac{425}{2}\\ =50\end{array}$

In tabular form:

  $\begin{array}{cc}\text{Secant}& \text{Slope}\\ P{Q}_1& 43\\ P{Q}_2& 46\\ P{Q}_3& 50\\ P{Q}_4& 50\end{array}$

Units for the slopes will be unit of distance/ unit of elapsed time. So, unit is: meters/second.

b.

To determine

To estimate: The velocity at the point P .

Answer to Problem 7E

The estimated velocity at point P is 50 meter/ sec.

Explanation of Solution

Given information:

  

The points are: $Q_1=\left(10,225\right),Q_2=\left(14,375\right),Q_3=\left(16.5,475\right),Q_4\left(18,550\right)\text{ and }P=\left(20,650\right)$

Concept used:

The slope of secant between the points $\left(a,f\left(a\right)\right)\text{ and }\left(b,f\left(b\right)\right)$ is given by: $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ and for the velocity at any point between $\left(a,f\left(a\right)\right)\text{ and }\left(b,f\left(b\right)\right)$ is the slope of the secant between the points.

Calculation:

Slope of the secant $P{Q}_4\text{ where }{Q}_4=\left(18,550\right)\text{ and }P=\left(20,650\right)$ is:

  $\begin{array}{c}P{Q}_4=\frac{650-550}{20-18}\\ =\frac{425}{2}\\ =50\end{array}$

Hence the estimated velocity is 50m/sec