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Chapter 1: Problem 62

Write a sentence using the variation terminology of this section to describe the formula. Surface area of a sphere: \(S=4 \pi r^{2}\)

Short Answer

Expert verified
The surface area of a sphere varies directly as the square of its radius.

Step by step solution

01

Understanding the Variation

First, understand the concept of variation. When we say one quantity varies with another, it means they change together in a specific way. In direct variation, when one quantity increases, the other also increases. In inverse variation, when one quantity increases, the other decreases.
02

Identifying the Variation in the Formula

Look at the formula \(S=4 \pi r^{2}\). Here, the surface area \(S\) is calculated based on the square of the radius \(r\). This means \(S\) and \(r^{2}\) change together. As \(r\) increases or decreases, \(S\) will also increase or decrease, respectively. As such, this is a direct variation relationship.
03

Writing the Sentence Using Variation Terminology

Finally, combining the characteristics of the variation relationship from the previous step, a sentence can be formulated to describe the formula. The sentence would be: The surface area of a sphere varies directly as the square of its radius.

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Most popular questions from this chapter

Write a sentence using the variation terminology of this section to describe the formula. Volume of a sphere: \(V=\frac{4}{3} \pi r^{3}\)

Restrict the domain of \(f(x)=x^{2}+1\) to \(x \geq 0 .\) Use a graphing utility to graph the function. Does the restricted function have an inverse function? Explain.

Find a mathematical model representing the statement. (In each case, determine the constant of proportionality.) \(A\) varies directly as \(r^{2} .(A=9 \pi\) when \(r=3 .)\)

Use the given value of \(k\) to complete the table for the direct variation model $$y=k x^{2}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=k x^{2} & & & & & \\ \hline \end{array}$$ $$ k=\frac{1}{2} $$

The total sales (in billions of dollars) for CocaCola Enterprises from 2000 through 2007 are listed below. (Source: Coca-Cola Enterprises, Inc.) $$\begin{array}{llll} 2000 & 14.750 & 2004 & 18.185 \\ 2001 & 15.700 & 2005 & 18.706 \\ 2002 & 16.899 & 2006 & 19.804 \\ 2003 & 17.330 & 2007 & 20.936 \end{array}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the total revenue (in billions of dollars) and let \(t=0\) represent 2000 . (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the sales of Coca-Cola Enterprises in 2008 . (f) Use your school's library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e).
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