Chapter 2

For the following exercises, use the model for the period of a pendulum, \(T\), such that \(T=2 \pi \sqrt{\frac{L}{g}}\), where the length of the pendulum is \(L\) and the acceleration due to gravity is g. If the gravity is \(32 \mathrm{ft} / \mathrm{s}^{2}\) and the period equals \(1 \mathrm{~s}\), find the length to the nearest in. (12 in. = \(1 \mathrm{ft}\) ). Round your answer to the nearest in.

To solve the quadratic equation \(x^{2}+5 x-7=4,\) we can graph these two equations $$\begin{array}{l} Y_{1}=x^{2}+5 x-7 \\ Y_{2}=4 \end{array}$$ and find the points of intersection. Recall \(2^{\text {nd }} \mathrm{CALC}\) 5:intersection. Do this and find the solutions to the nearest tenth.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{1}{i}+\frac{4}{i^{3}} $$

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as \(Y 1,\) then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values. $$ 0.537 x-2.19 y=100 $$

For the following exercises, write the interval in set-builder notation. \((4, \infty)\)

For the following exercises, use a model for body surface area, BSA, such that \(B S A=\sqrt{\frac{w h}{3600}}\), where w= weight in kg and \(h=\) height in cm. Find the height of a \(72-\mathrm{kg}\) female to the nearest \(\mathrm{cm}\) whos \(B S A=1.8\).

To solve the quadratic equation \(0.3 x^{2}+2 x-4=2,\) we can graph these two equations $$\begin{array}{l} Y_{1}=0.3 x^{2}+2 x-4 \\ Y_{2}=2 \end{array}$$ and find the points of intersection. Recall \(2^{\text {nd }}\) CALC 5:intersection. Do this and find the solutions to the nearest tenth.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{1}{i^{11}}-\frac{1}{i^{21}} $$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. The volume formula for a cylinder is \(V=\pi r^{2} h .\) Using the symbol \(\pi\) in your answer, find the volume of a cylinder with a radius, \(r\), of \(4 \mathrm{~cm}\) and a height of \(14 \mathrm{~cm}\).

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as \(Y 1,\) then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values. $$ 4,500 x-200 y=9,528 $$

For the following exercises, use your graphing calculator to input the linear graphs in the \(Y=\) graph menu. After graphing it, use the \(2^{\text {nd }}\) CALC button and 1:value button, hit enter. At the lower part of the screen you will see " \(x=\) " and a blinking cursor. You may enter any number for \(x\) and it will display the y value for any \(x\) value you input. Use this and plug in \(x=0\), thus finding the \(y\) -intercept, for each of the following graphs. \(Y_{1}=-2 x+5\)

For the following exercises, write the interval in set-builder notation. [-3,5)

For the following exercises, use a model for body surface area, BSA, such that \(B S A=\sqrt{\frac{w h}{3600}}\), where w= weight in kg and \(h=\) height in cm. Find the weight of a \(177-\mathrm{cm}\) male to the nearest \(\mathrm{kg}\) whose \(B S A=2.1\).

Beginning with the general form of a quadratic equation, \(a x^{2}+b x+c=0,\) solve for \(x\) by using the completing the square method, thus deriving the quadratic formula.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ i^{7}\left(1+i^{2}\right) $$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(h: V=\pi r^{2} h\)

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as \(Y 1,\) then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values. $$ \frac{200-30 y}{x}=70 $$

For the following exercises, use your graphing calculator to input the linear graphs in the \(Y=\) graph menu. After graphing it, use the \(2^{\text {nd }}\) CALC button and 1:value button, hit enter. At the lower part of the screen you will see " \(x=\) " and a blinking cursor. You may enter any number for \(x\) and it will display the y value for any \(x\) value you input. Use this and plug in \(x=0\), thus finding the \(y\) -intercept, for each of the following graphs. \(\mathrm{Y}_{1}=\frac{3 x-8}{4}\)

For the following exercises, write the interval in set-builder notation. [-4,1]\(\cup[9, \infty)\)

Show that the sum of the two solutions to the quadratic equation is \(-\frac{b}{a}\).

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ i^{-3}+5 i^{7} $$

Starting with the point-slope formula \(y-y_{1}=m\left(x-x_{1}\right),\) solve this expression for \(x\) in terms of \(x_{1}, y, y_{1},\) and \(m\).

For the following exercises, use your graphing calculator to input the linear graphs in the \(Y=\) graph menu. After graphing it, use the \(2^{\text {nd }}\) CALC button and 1:value button, hit enter. At the lower part of the screen you will see " \(x=\) " and a blinking cursor. You may enter any number for \(x\) and it will display the y value for any \(x\) value you input. Use this and plug in \(x=0\), thus finding the \(y\) -intercept, for each of the following graphs. \(\mathrm{Y}_{1}=\frac{x+5}{2}\)

A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 \(\mathrm{ft} .^{2} .\) Solve the quadratic equation to find the length and width.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{(2+i)(4-2 i)}{(1+i)} $$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve for \(r: V=\pi r^{2} h\)

Starting with the standard form of an equation \(A x+B y=C\) solve this expression for \(y\) in terms of \(A, B, C\) and \(x\). Then put the expression in slopeintercept form.

For the following exercises, use your graphing calculator to input the linear graphs in the \(Y=\) graph menu. After graphing it, use the \(2^{\text {nd }}\) CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see "left bound?" and a blinking cursor on the graph of the line. Move this cursor to the left of the \(x\) -intercept, hit ENTER. Now it says "right bound?" Move the cursor to the right of the x-intercept, hit ENTER. Now it says "guess?" Move your cursor to the left somewhere in between the left and right bound near the \(x\) -intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the \(x\) -intercept or the "zero" to the \(y\) -value. Use this to find the \(x\) -intercept. Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries. $$ \mathrm{Y}_{1}=-8 x+6 $$

Abercrombie and Fitch stock had a price given as \(P=0.2 t^{2}-5.6 t+50.2,\) where \(t\) is the time in months from 1999 to 2001. \((t=1\) is January 1999). Find the two months in which the price of the stock was $$\$ 30$$.

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{(1+3 i)(2-4 i)}{(1+2 i)} $$

For the following exercises, use your graphing calculator to input the linear graphs in the \(Y=\) graph menu. After graphing it, use the \(2^{\text {nd }}\) CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see "left bound?" and a blinking cursor on the graph of the line. Move this cursor to the left of the \(x\) -intercept, hit ENTER. Now it says "right bound?" Move the cursor to the right of the x-intercept, hit ENTER. Now it says "guess?" Move your cursor to the left somewhere in between the left and right bound near the \(x\) -intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the \(x\) -intercept or the "zero" to the \(y\) -value. Use this to find the \(x\) -intercept. Note: With linear/straight line functions the zero is not really a "guess," but it is necessary to enter a "guess" so it will search and find the exact \(x\) -intercept between your right and left boundaries. With other types of functions (more than one \(x\) -intercept), they may be irrational numbers so "guess" is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries. $$ \mathrm{Y}_{1}=4 x-7 $$

Suppose that an equation is given \(p=-2 x^{2}+280 x-1000\) where \(x\) represents the number of items sold at an auction and \(p\) is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using \(2^{\text {nd }}\) CALC maximum. To obtain a good window for the curve, set \(x[0,200]\) and \(y\) \([0,10000].\)

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{(3+i)^{2}}{(1+2 i)^{2}} $$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. The formula for the circumference of a circle is \(C=2 \pi r .\) Find the circumference of a circle with a diameter of 12 in. (diameter = \(2 r)\). Use the symbol \(\pi\) in your final answer.

Given that the following coordinates are the vertices of a rectangle, prove that this truly is a rectangle by showing the slopes of the sides that meet are perpendicular. (-1,1),(2,0),(3,3) and (0,4).

For the following exercises, input the left-hand side of the inequality as a \(Y 1\) graph in your graphing utility. Enter \(y 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, \(1: a b s(.\) Find the points of intersection, recall \(\left(2^{\text {nd }}\right.\) CALC 5 :intersection, \(1^{\text {st }}\) curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x+2|-5<2 $$

A formula for the normal systolic blood pressure for a man age \(A,\) measured in \(\mathrm{mmHg}\), is given as \(P=0.006 A^{2}-0.02 A+120\) Find the age to the nearest year of a man whose normal blood pressure measures \(125 \mathrm{mmHg}\).

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{3+2 i}{2+i}+(4+3 i) $$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question. Solve the formula from the previous question for \(\pi\). Notice why \(\pi\) is sometimes defined as the ratio of the circumference to its diameter.

A man drove 10 mi directly east from his home, made a left turn at an intersection, and then traveled 5 mi north to his place of work. If a road was made directly from his home to his place of work, what would its distance be to the nearest tenth of a mile?

For the following exercises, input the left-hand side of the inequality as a \(Y 1\) graph in your graphing utility. Enter \(y 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, \(1: a b s(.\) Find the points of intersection, recall \(\left(2^{\text {nd }}\right.\) CALC 5 :intersection, \(1^{\text {st }}\) curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ \frac{-1}{2}|x+2|<4 $$

The cost function for a certain company is \(C=60 x+300\) and the revenue is given by \(R=100 x-0.5 x^{2}\). Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of \(x\) (production level) that will create a profit of $$\$ 300.$$

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{4+i}{i}+\frac{3-4 i}{1-i} $$

The slope for a wheelchair ramp for a home has to be \(\frac{1}{12}\). If the vertical distance from the ground to the door bottom is \(2.5 \mathrm{ft}\), find the distance the ramp has to extend from the home in order to comply with the needed slope.

For the following exercises, input the left-hand side of the inequality as a \(Y 1\) graph in your graphing utility. Enter \(y 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, \(1: a b s(.\) Find the points of intersection, recall \(\left(2^{\text {nd }}\right.\) CALC 5 :intersection, \(1^{\text {st }}\) curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |4 x+1|-3>2 $$

A falling object travels a distance given by the formula \(d=5 t+16 t^{2} \mathrm{ft},\) where \(t\) is measured in seconds. How long will it take for the object to travel \(74 \mathrm{ft}\) ?

For the following exercises, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{3+2 i}{1+2 i}-\frac{2-3 i}{3+i} $$

If the profit equation for a small business selling \(x\) number of item one and \(y\) number of item two is \(p=3 x+4 y,\) find the \(y\) value when \(p=\ 453\) and \(x=75\).

Given these four points: \(A(1,3)\) \(B(-3,5), C(4,7),\) and \(D(5,-4)\) find the coordinates of the midpoint of line segments \(\overline{\mathrm{AB}}\) and \(\overline{\mathrm{CD}} .\)

For the following exercises, input the left-hand side of the inequality as a \(Y 1\) graph in your graphing utility. Enter \(y 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, \(1: a b s(.\) Find the points of intersection, recall \(\left(2^{\text {nd }}\right.\) CALC 5 :intersection, \(1^{\text {st }}\) curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ |x-4|<3 $$