Chapter 3.1, Problem 30E

(a)

To determine

To find: The sketch for the graph of the elevation ( y ) as the function of the distance downriver ( x ).

Answer to Problem 30E

The sketch of the graph is shown in Figure 1.

Explanation of Solution

Given Data:

The given table is shown in Table 1.

Table 1

Distance down river in miles	River Elevation Feet
0.00	1577
0.56	1512
0.92	1448
1.19	1384
1.30	1319
1.39	1255
1.57	1191
1.74	1126
1.98	1062
2.18	998
2.41	933
2.64	869
3.24	805

Calculation:

From the given table the graph of the data is shown in Figure 1.

  

Figure 1

In the above figure the x is the distance down the river and the y is the river elevation.

(b)

To determine

To find: The approximate graph of the given data.

Answer to Problem 30E

The required graph is shown in Figure 2.

Explanation of Solution

Consider the formula for the midpoint is,

  $m=\frac{\text{initial}\text{distance}+\text{next}\text{distance}}{2}$

The formula for the slope is,

  $\frac{\Delta y}{\Delta x}=\frac{\text{initial}\text{distance}\text{+next}\text{distance}}{\text{2}}$

Then, form the table 1 the table for the mid-point interval and the slope is shown in Table 2

Table 2

Midpoint of the interval	Slope
0.28	-116.07
0.74	-177.78
1.055	-237.04
1.245	-590.91
1.345	-711.11
1.48	-355.56
1.655	-382.35
1.86	-266.67
2.08	-320.00
2.295	-282.61
2.525	-278.26
2.94	-106.67

From the above data the approximate graph is shown in Figure 2

  

Figure 2

(c)

To determine

To find: The unit measure of the gradient.

Answer to Problem 30E

The unit of the gradient is in feet per mile.

Explanation of Solution

Consider the formula for the gradient is,

  $G=\frac{\text{average}\text{change}\text{in}\text{elevation}}{\text{given}\text{distance}}$

Since, the value of the elevation is feet and the distance down the river is in miles the unit of the gradient is in feet per mile.

(d)

To determine

To find: The unit measure that is appropriate for the derivative.

Answer to Problem 30E

The unit of derivative are feet per mile.

Explanation of Solution

The derivative is given by,

  $\frac{dy}{dx}$

Here, y is the elevation that is in feet.

The distance is $x$ down river in miles.

The unit of derivative are feet per mile.

(e)

To determine

To find: The identification of the most dangerous section of the river by analysing the graph of part (a)

Answer to Problem 30E

The points where the change is the river elevation causes rapid increase in the speed of the river flow and this is the most dangerous section of the river.

Explanation of Solution

The steepest part of the curve and in this way the elevation drops rapidly and the most likely location for significant rapids. The points where the change is the river elevation causes rapid increase in the speed of the river flow and this is the most dangerous section of the river.

(f)

To determine

To find: The way in which the most dangerous section of the river is identified by the analysing the graph in ( b ) Explain.

Answer to Problem 30E

The lowest point is the point where there is signification change in depth and hence it is the most dangerous section of the river.

Explanation of Solution

The lowest point on the graph is the point where the elevation drops most rapidly and therefore the most likely location of the significant rapids.

Thus, the lowest point is the point where there is signification change in depth and hence it is the most dangerous section of the river.