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Chapter 3: Problem 52

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Using the language of variation, I can now state the formula for the area of a trapezoid, \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right),\) as, "A trapezoid's area varies jointly with its height and the sum of its bases."

Short Answer

Expert verified
The statement 'A trapezoid's area varies jointly with its height and the sum of its bases.' makes sense because this statement accurately represents the mathematical relationship expressed in the formula for the area of a trapezoid.

Step by step solution

01

Understand the language of variation

In mathematics, 'variation' explains how one quantity changes in relation to another. In 'joint variation', the variation of a quantity is directly with the multiplication of two or more other quantities. In this instance, you're looking at the area of a trapezoid, and checking whether it varies jointly with its height and the sum of its bases.
02

Relating area of trapezoid to variation

The formula for the area of a trapezoid is \(A=\frac{1}{2} h\left(b_{1}+b_{2}\right)\), where \(A\) is the area, \(h\) is the height, and \(b_{1}\) and \(b_{2}\) are the lengths of the bases. Clearly, the area (\(A\)) is obtained by multiplying the height (\(h\)) by the sum of the bases (\(b_{1} + b_{2}\)). Thus by definition, it can be stated that 'A trapezoid's area varies jointly with its height and the sum of its bases.'
03

Confirming joint variation

From the previous step, you can confirm that the statement 'A trapezoid's area varies jointly with its height and the sum of its bases.' is indeed valid, as it correctly represents the relationship between the area of a trapezoid, its height, and the sum of its bases.

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Most popular questions from this chapter

The functions $$f(x)=0.0875 x^{2}-0.4 x+66.6$$ and $$g(x)=0.0875 x^{2}+1.9 x+11.6$$ model a car's stopping distance, \(f(x)\) or \(g(x),\) in feet, traveling at \(x\) miles per hour. Function \(f\) models stopping distance on dry pavement and function g models stopping distance on wet pavement. The graphs of these functions are shown for \(\\{x | x \geq 30\\} .\) Notice that the figure does not specify which graph is the model for dry roads and which is the model for wet roads. Use this information to solve. (GRAPH CANNOT COPY). a. Use the given functions to find the stopping distance on dry pavement and the stopping distance on wet pavement for a car traveling at 35 miles per hour. Round to the nearest foot. b. Based on your answers to part (a), which rectangular coordinate graph shows stopping distances on dry pavement and which shows stopping distances on wet pavement? c. How well do your answers to part (a) model the actual stopping distances shown in Figure 3.43 on page \(411 ?\) d. Determine speeds on dry pavement requiring stopping distances that exceed the length of one and one-half football fields, or 540 feet. Round to the nearest mile per hour. How is this shown on the appropriate graph of the models?

Solve each inequality using a graphing utility. $$ \frac{x-4}{x-1} \leq 0 $$

Use the four-step procedure for solving variation problems given on page 424 to solve. If all men had identical body types, their weight would vary directly as the cube of their height. Shown below is Robert Wadlow, who reached a record height of 8 feet 11 inches \((107 \text { inches ) before his death at age } 22 .\) If a man who is 5 feet 10 inches tall \((70\) inches) with the same body type as Mr. Wadlow weighs 170 pounds, what was Robert Wadlow's weight shortly before his death?

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\).

What is a rational function?
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