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nature of mathematics

Questions and Answers of Nature Of Mathematics

If \(A x^{2}+C y^{2}+D x+E y+F=0\), for Problems 9-13, then list conditions on the constants to assure that the indicated graph results.a line \(A=C=0\)
Which of the sets in Problems 3-14 are functions?\(\{69,82,44,37\}\)
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y
Tell what the output value is for each of the function machines in Problems 15-20 for(a) 4 ,(b) 6 ,(c) -8 ,(d) \(\frac{1}{2}\),(e) \(t\) f machine Add 5- Multiply by 2,
Tell what the output value is for each of the function machines in Problems 15-20 for(a) 4 ,(b) 6 ,(c) -8 ,(d) \(\frac{1}{2}\),(e) \(t\) g machine Add 8- Multiply
Sketch the graph of each equation in Problems 3-30.\(x=\frac{1}{2} y^{2}\)
Tell what the output value is for each of the function machines in Problems 15-20 for(a) 4 ,(b) 6 ,(c) -8 ,(d) \(\frac{1}{2}\),(e) \(t\) 2+1 M x+1
Find the value of each function in Problems 14-17.\(f(x)=3 x+2\); find \(f(6) .\)
Sketch the requested conic sections in Problems 14-23 using the definition.A circle with radius 3
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y>5 x-3\)
Sketch the graph of each equation in Problems 3-30.\(y=2 x^{3}\)
Find the value of each function in Problems 14-17.\(g(x)=x^{2}-3\); find \(g(0) .\)
Tell what the output value is for each of the function machines in Problems 15-20 for(a) 4 ,(b) 6 ,(c) -8 ,(d) \(\frac{1}{2}\),(e) \(t\) 3x S X+X+X O +X+X __xx+x+x_x+x+
Sketch the requested conic sections in Problems 14-23 using the definition.A circle with radius 5
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y>5 x-3\)
Sketch the graph of each equation in Problems 3-30.\(y=-x^{3}\)
Find the value of each function in Problems 14-17.\(F(x)=5 x+25\); find \(F(10) .\)
Sketch the requested conic sections in Problems 14-23 using the definition.A parabola with the distance between the directrix and focus 2 units
Tell what the output value is for each of the function machines in Problems 15-20 for(a) 4 ,(b) 6 ,(c) -8 ,(d) \(\frac{1}{2}\),(e) \(t\) *L Y-Y 2222 2x 2x-1 50 75 2x-1
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y>-2 x+3\)
Sketch the graph of each equation in Problems 3-30.\(x=-y^{3}\)
Find the value of each function in Problems 14-17.\(m(x)=5\); find \(m(10)\).
Sketch the requested conic sections in Problems 14-23 using the definition.A parabola with the distance between the directrix and focus 4 units
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y>-2 x+3\)
Sketch the graph of each equation in Problems 3-30.\(y=\frac{1}{3} x^{2}\)
An amount of money \(A\) results from investing a sum \(P\) at a simple interest rate of \(7 \%\) for 10 years, and is specified by the formula\[A=1.7 P\]Graph this equation, where \(P\) is the
Sketch the requested conic sections in Problems 14-23 using the definition.A parabola with the distance between the directrix and focus 6 units
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y>3 x-3\)
Sketch the graph of each equation in Problems 3-30.\(y=-\frac{2}{3} x^{2}\)
If a golf ball is hit with an initial velocity of 128 feet per second, its height can be calculated according to the formula\[y=128 t-16 t^{2}\]where \(t\) is the length of time (in seconds) after
Tell what the output value is for each of the function machines in Problems 15-20 for(a) 4 ,(b) 6 ,(c) -8 ,(d) \(\frac{1}{2}\),(e) \(t\) x 421 + N/A C x +
Sketch the requested conic sections in Problems 14-23 using the definition.A parabola with the distance between the directrix and focus 1 unit
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y>3 x-3\)
Sketch the graph of each equation in Problems 3-30.\(y=\frac{1}{10} x^{2}\)
Draw a population curve for a city whose growth rate is \(1.3 \%\) and whose present population is 53,000 . The equation is\[P=P_{0} e^{r t}\]Let \(t=0,10, \cdots, 50\) to help you find points for
Sketch the requested conic sections in Problems 14-23 using the definition.An ellipse with the distance between the foci 10 units and the sum of the distances 12 units
If \(f(x)=x-7\), finda. \(f(15)\)b. \(f(-9)\)c. \(f(p)\)
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y \leq-2 x+5\)
Sketch the graph of each equation in Problems 3-30.\(y=-\frac{1}{10} x^{2}\)
Sketch the requested conic sections in Problems 14-23 using the definition.An ellipse with the distance between the foci 10 units and the sum of the distances 14 units
If \(g(x)=2 x\), finda. \(g(100)\)b. \(g(-25)\)c. \(g(m)\)
Graph the first-degree inequalities in two unknowns in Problems 13-48.\(y
Sketch the graph of each equation in Problems 3-30.\(y=\frac{1}{1,000} x^{2}\)
Sketch the requested conic sections in Problems 14-23 using the definition.A hyperbola with the distance between the foci 10 units and the difference of the distances 8 units
A radar antenna is constructed so that a cross section along its axis is a parabola with the receiver at the focus. Find the focus if the antenna is \(12 \mathrm{~m}\) across and its depth is \(4
Beams of light parallel to the axis of the parabolic mirror shown in Figure 15.44 strike the mirror and are reflected. Find the distance from the vertex to the point where the beams
A parabolic archway has the dimensions shown in Figure 15.42.If we draw coordinate axes as shown, it can be shown that the equation of the archway portion is \[x^{2}=-4 c(y+3)\]for some constant
Your test scores are 78, 95, 87, and 92.What score on the fifth exam will make your average score at least 90 ?
You can rent a car for \(\$ 50 /\) day \(+55 \% /\) mile (option A) or \(\$ 89 /\) day \(+25 otin /\) mile (option B). For how many miles will option A be cheaper than option B?
The population of Texas was 17.0 million in 1990 and 19.8 million in 1998. Let \(x\) be the year (let 1990 be the base year; that is, \(x=0\) represents the year 1990 and \(x=8\) represents 1998) and
From a square whose side has length \(x\), measured in inches, create a new square whose side is 5 in. longer. Find an expression for the difference between the areas of the two squares as a function
John is given the option of a straight salary of \(\$ 2,500 /\) week (no commission option), or a salary of \(\$ 500 /\) week plus \(10 \%\) of the weekly sales. How many sales are necessary for the
The equation for the standard normal curve (the normal curve with mean 0 and standard deviation 1) graphs as an exponential curve. Graph this curve, whose equation is\[y=\frac{e^{-x^{2} / 2}}{\sqrt{2
Show that the equation of a line passing through \((h, k)\) with slope \(m\) is\[y-k=m(x-h)\]This is called the point-slope form.
From a square whose side has length \(x\), measured in meters, create a new square whose side is \(10 \mathrm{~m}\) longer. Find an expression for the sum of the areas of the two squares as a
Jerry is given the option of a straight salary of \(\$ 4,600 /\) week (no commission option), or a salary of \(\$ 1,000 /\) week plus \(15 \%\) of the weekly sales. How many sales are necessary for
A parabolic reflector (see Figure 15.43) is constructed so that a cross section along its axis is a parabola with the light source at the focus. Find the location of the focus if the reflector is
Graph the curve \(y=2^{-x^{2}}\) and compare with the standard normal curve (Problem 59). List some similarities and some differences.
Begin with the point-slope form in Problem 58 and derive the two-point form. If the line passes through \(\left(x_{1}, y_{1}ight)\) and \(\left(x_{2}, y_{2}ight)\),
Find the area of a square as a function of its perimeter.
Suppose you want to invest \(\$ 30,000\). Part of this investment will be in a guaranteed rate \(5 \%\) annuity, and the remainder will be invested in a "friend's" business and she will pay you "a
Show that the equation of the line with \(x\)-intercept \(a\) and \(y\)-intercept \(b\) is\[\frac{x}{a}+\frac{y}{b}=1\]\((a eq 0, b eq 0)\). This is called the intercept form.
Find the area of a circle as a function of its circumference.
Five identical containers (shoe boxes, paper cups, etc.) must be prepared for this problem, with contents as follows: There are five boxes containing red and white items (such as marbles, poker
Will a baseball player's batting average always change when the player is at bat? That is, under what conditions will the batting average stay the same? Let \(a\) be the number of times at bat, and
A stone tunnel is to be constructed such that the opening is a semielliptic arch, as shown in Figure 15.45. It is necessary to know the height at 4 -ft intervals from the center. That is, how high is
In Problems 1-5, perform the indicated matrix operations, if possible, where \[\begin{array}{ll}{[\mathrm{A}]=\left[\begin{array}{rr}1 & 0 \\2 & -1\end{array}ight]} &
When are two matrices equal?
Suppose your group conducts an experiment at a local department store. You walk up a rising escalator and you take one step per second to reach the top in 20 seconds. Next, you walk up the same
A box contains twenty-five more dimes than nickels. How many of each type of coin is there if the total value is \(\$ 7.15\) ?
In the Boston Museum of Fine Arts is a display of carefully made stone cubes found in the ruins of Mohenjo-Daro of the Indus. The stones are a set of weights that exhibit the binary pattern,
Can you relate to the Fox Trot comic strip at the beginning of this section? Why or why not? Why might you need to solve a system of equations?
What is a matrix?
In Problems 1-5, perform the indicated matrix operations, if possible, where \[\begin{array}{ll}{[\mathrm{A}]=\left[\begin{array}{rr}1 & 0 \\2 & -1\end{array}ight]} &
Describe the process for adding matrices.
Two ranchers sold a herd of cattle and received as many dollars for each animal as there were cattle in the herd. With the money, they bought a flock of sheep at \(\$ 10\) a head and then a lamb with
A box contains only dimes and quarters. The number of dimes is three less than twice the number of quarters. If the total value of the coins is \(\$ 22.65\), how many dimes are in the box?
What is a system of linear equations?
Describe a procedure for solving a system of linear inequalities.
List the elementary row operations and briefly describe each.
In Problems 1-5, perform the indicated matrix operations, if possible, where \[\begin{array}{ll}{[\mathrm{A}]=\left[\begin{array}{rr}1 & 0 \\2 & -1\end{array}ight]} &
Describe the process for multiplying matrices.
Suppose your group has just been hired by a company called Alco. You are asked to analyze its operations and make some recommendations about how it can comply at a minimum cost with recent orders of
Forty-two coins have a total value of \(\$ 6.90\). If the coins are all nickels and quarters, how many are nickels?
Read the "What in the World?" at the beginning of this chapter. Rank teams A, B, C, D, E, and F if you are given the following information. Team A beats F, and ties \(\mathrm{C}\) and \(\mathrm{D}\);
Distinguish inconsistent and dependent systems.
What is a linear programming problem?
What is Gauss-Jordan elimination?
In Problems 1-5, perform the indicated matrix operations, if possible, where \[\begin{array}{ll}{[\mathrm{A}]=\left[\begin{array}{rr}1 & 0 \\2 & -1\end{array}ight]} &
What is the inverse of a matrix?
A boat travels \(25 \mathrm{mph}\) relative to the riverbank while going downstream and only \(15 \mathrm{mph}\) returning upstream. What is the boat's speed in still water?
Knot Theory Get a piece of string with two free ends, and tie those ends together with a knot. Some knots that you can tie will hold the ends of the string together and other knots will not (no pun
Describe the graphing method for solving a system of equations.
What is an objective function? What do we mean by constraints?
In Problems 4-13, decide whether the statement is true or false. If it is false, tell what is wrong.Entry \(a_{34}\) is in row 3 , column 4 .
In Problems 1-5, perform the indicated matrix operations, if possible, where \[\begin{array}{ll}{[\mathrm{A}]=\left[\begin{array}{rr}1 & 0 \\2 & -1\end{array}ight]} &
What is a nonsingular matrix?
An airliner flies at \(510 \mathrm{mph}\) in the jet stream, but only \(270 \mathrm{mph}\) returning against it. What is the speed of the plane in still air?
Describe the substitution method for solving a system of equations.
Graph the solution of each system given in Problems 5-18.a. \(\left\{\begin{array}{l}x \geq 0 \\ y \leq 0\end{array}ight.\)b. \(\left\{\begin{array}{l}x \geq 0 \\ y \geq 0\end{array}ight.\)

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