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Chapter 3: Problem 12

An overflow pipeline: An overflow pipeline for a pond is to run in a straight line from the pond at maximum water level a distance of 96 horizontal feet to a drainage area that is 5 vertical feet below the maximum water level (see Figure 3.36). How much lower is the pipe at the end of each 12 -foot horizontal stretch?

Short Answer

Expert verified
The pipe drops \( \frac{5}{8} \) foot at the end of each 12-foot horizontal stretch.

Step by step solution

01

Understand the given triangle

The problem describes a right triangle where one leg is the horizontal distance, 96 feet, and the other leg is the vertical drop, 5 feet. The hypotenuse is the route of the pipeline.
02

Find the slope of the pipeline

The slope of a line in the context of a right triangle is given by the ratio of the change in vertical distance (rise) to the change in horizontal distance (run). Here, the slope of the pipeline is \( \frac{5}{96} \).
03

Calculate the vertical drop for each 12-foot horizontal segment

Since the slope is \( \frac{5}{96} \), in each 12-foot horizontal segment, the vertical drop can be calculated using the formula: \( \text{Vertical drop} = \frac{5}{96} \times 12 \).
04

Perform the calculation

Calculate the vertical drop using the formula from the previous step: \[ \text{Vertical drop} = \frac{5}{96} \times 12 = \frac{60}{96} = \frac{5}{8} \text{ feet} \].
05

Check consistency with total drop

There are \( 96 / 12 = 8 \) segments. If each segment has a vertical drop of \( \frac{5}{8} \), the total vertical drop is \( 8 \times \frac{5}{8} = 5 \text{ feet} \), which checks out as consistent with the problem statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Calculation
The concept of slope comes into play when trying to understand how steep a line or a segment is. In math, particularly in geometry and trigonometry, the slope of a straight line is a measure of its inclination. It represents the ratio of the vertical drop to the horizontal distance, often termed as rise over run.
In the context of a right triangle formed by an overflow pipeline, the slope can be calculated by determining how much the vertical level decreases over a certain horizontal distance. It is calculated using the formula:\[ \text{slope} = \frac{\text{rise}}{\text{run}} \]So when given a vertical drop of 5 feet over a total horizontal stretch of 96 feet, the slope would be \[ \frac{5}{96} \].
Understanding the slope helps us figure out how much the pipe will drop over a smaller horizontal segment. For example, if you want to calculate the drop over each 12-foot segment, this slope ratio aids in predicting how much lower the end of each segment should be.
Right Triangle
A right triangle is a crucial part of this exercise. It is a shape that has one angle exactly equal to 90 degrees, often depicted as an L shape. This type of triangle is incredibly useful in trigonometry because we can apply simple mathematical theories to solve various problems.
In our pipeline problem, the triangle is formed by:
  • The horizontal distance of 96 feet.
  • The vertical drop of 5 feet.
  • The pipeline itself, which is the hypotenuse of the right triangle.
Understanding this geometry allows us to apply the Pythagorean theorem and slope calculations effectively. Right triangles also enable the use of trigonometric ratios like sine, cosine, and tangent, which could be used in other contexts but aren't necessary for this specific problem.
Vertical Drop Calculation
Calculating the vertical drop is an essential part of determining how the pipeline changes in height over a specified distance.
When you have a known slope, you can determine how much the height changes per unit of horizontal distance. The formula to find the vertical drop for a specific span is:\[ \text{Vertical drop} = \text{slope} \times \text{horizontal distance} \]In this problem, the slope is \[ \frac{5}{96} \], and for each 12-foot horizontal segment, the vertical drop can be determined by:\[ \frac{5}{96} \times 12 = \frac{5}{8} \text{ feet} \].
This ensures every 12-foot stretch of the pipeline results in a vertical drop of 0.625 feet. Such calculations are useful for understanding infrastructural projects, ensuring water flows correctly from one point to another.

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Most popular questions from this chapter

Is a linear model appropriate? The number, in thousands, of bacteria in a petri dish is given by the table below. Time is measured in hours. $$ \begin{array}{|c|c|} \hline \begin{array}{l} \text { Time in hours since } \\ \text { experiment began } \end{array} & \begin{array}{c} \text { Number of bacteria } \\ \text { in thousands } \end{array} \\ \hline 0 & 1.2 \\ \hline 1 & 2.4 \\ \hline 2 & 4.8 \\ \hline 3 & 9.6 \\ \hline 4 & 19.2 \\ \hline 5 & 38.4 \\ \hline 6 & 76.8 \\ \hline \end{array} $$ The table below shows enrollment, \({ }^{18}\) in millions of people, in public colleges in the United States during the years from 2001 through 2005 . $$ \begin{array}{|c|c|} \hline \text { Date } & \text { Enrollment in millions } \\ \hline 2001 & 12.23 \\ \hline 2002 & 12.75 \\ \hline 2003 & 12.86 \\ \hline 2004 & 12.98 \\ \hline 2005 & 13.02 \\ \hline \end{array} $$ a. Plot the data points for number of bacteria. Does it look reasonable to approximate these data with a straight line? b. Plot the data points for college enrollment. Does it look reasonable to approximate these data with a straight line?

Gross national product: The United States gross national product, in trillions of dollars, is given in the table below. $$ \begin{array}{|c|c|} \hline \text { Date } & \text { Gross national product } \\ \hline 2002 & 10.50 \\ \hline 2003 & 11.02 \\ \hline 2004 & 11.76 \\ \hline 2005 & 12.49 \\ \hline 2006 & 13.28 \\ \hline \end{array} $$ a. Find the equation of the regression line, and explain the meaning of its slope. (Round regression line parameters to two decimal places.) b. Plot the data points and the regression line. c. Suppose that in 2006 a prominent economist predicted that by 2012 , the gross national product would reach 18 trillion dollars. Does your information from part a support that conclusion? If not, when would you predict that a gross national product of 18 trillion dollars would be reached?

Where lines with different slopes meet: On the same coordinate axes, draw one line with vertical intercept 2 and slope 3 and another with vertical intercept 4 and slope 1. Do these lines cross? If so, do they cross to the right or left of the vertical axis? In general, if one line has its vertical intercept below the vertical intercept of another, what conditions on the slope will ensure that the lines cross to the right of the vertical axis?

The effect of sampling error on linear regression: A stream that feeds a lake is flooding, and during this flooding period the depth of water in the lake is increasing. The actual depth of the water at a certain point in the lake is given by the linear function \(D=0.8 t+52\) feet, where \(t\) is measured in hours since the flooding began. A hydrologist does not have this function available and is trying to determine experimentally how the water level is rising. She sits in a boat and, each half-hour, drops a weighted line into the water to measure the depth to the bottom. The motion of the boat and the waves at the surface make exact measurement impossible. Her compiled data are given in the following table. $$ \begin{array}{|c|c|} \hline \begin{array}{l} t=\text { hours since } \\ \text { flooding began } \end{array} & \begin{array}{c} D=\text { measured } \\ \text { depth in feet } \end{array} \\ \hline 0 & 51.9 \\ \hline 0.5 & 52.5 \\ \hline 1 & 52.9 \\ \hline 1.5 & 53.3 \\ \hline 2 & 53.7 \\ \hline \end{array} $$ $$ \text { a. Plot the data points. } $$ b. Find the equation of the regression line for \(D\) as a function of \(t\), and explain in practical terms the meaning of the slope. c. Add the graph of the regression line to the plot of the data points. d. Add the graph of the depth function \(D=\) \(0.8 t+52\) to the picture. Does it appear that the hydrologist was able to use her data to make a close approximation of the depth function? e. What was the actual depth of the water at \(t=3\) hours? f. What prediction would the hydrologist's regression line give for the depth of the water at \(t=3\) ?

Growth in height: Between the ages of 7 and 11 years, a certain boy grows 2 inches taller each year. At age 9 he is 48 inches tall. a. Explain why, during this period, the function giving the height of the boy in terms of his age is linear. Identify the slope of this function. b. Use a formula to express the height of the boy as a linear function of his age during this period. Be sure to identify what the letters that you use mean. c. What is the initial value of the function you found in part \(b\) ? d. Studying a graph of the boy's height as a function of his age from birth to age 7 reveals that the graph is increasing and concave down. Does this indicate that his actual height (or length) at birth was larger or smaller than your answer to part c? Be sure to explain your reasoning.
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