Chapter 2

What is the total distance that two people travel in 3 \(\mathrm{h}\) if one of them is riding a bike at 15 \(\mathrm{mi} / \mathrm{h}\) and the other is walking at 3 \(\mathrm{mi} / \mathrm{h}\) ?

For the following exercises, use a calculator to help answer the questions. Evaluate \((l+i)^{k}-(l-i)^{k}\) for \(k=4,8\), and 12 . Predict the value for \(k=16\).

enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using \(2^{\text { nd }}\) CALC \(2 :\) zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero, enter), then right bound (move your cursor to the right of the zero, enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth. $$\mathrm{Y}_{1}=4 x^{2}+3 x-2$$

For the following exercises, write the set in interval notation. $$ \\{x | x < 4\\} $$

For the following exercises, solve for the unknown variable. $$ \left|x^{2}+2 x-36\right|=12 $$

Write the set in interval notation. $$ \\{x \mid x<4\\} $$

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular. \((-1,3)\) and \((5,1)\) \((-2,3)\) and \((0,9)\)

If the area model for a triangle is \(A=\frac{1}{2} b h,\) find the area of a triangle with a height of 16 in. and a base of 11 in.

Enter the expressions into your graphing utility and find the zeroes to the equation (the \(x\) -intercepts) by using \(2^{\text {nd }}\) CALC 2 :zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero, enter), then right bound (move your cursor to the right of the zero, enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth. $$ \mathrm{Y}_{1}=-3 x^{2}+8 x-1 $$

For each of the following exercises, use the graph in the figure below. Find the distance that \((-3,4)\) is from the origin.

For the following exercises, write the set in interval notation. $$ \\{x | x \text { is all real numbers }\\} $$

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 acceleration due to gravity is \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) and the period equals \(1 \mathrm{~s}, \mathrm{fi}\) d the length to the nearest \(\mathrm{cm}\) \((100 \mathrm{~cm}=1 \mathrm{~m})\)

Write the set in interval notation. $$ \\{x \mid x \text { is all real numbers }\\} $$

For the following exercises, find the slope of the lines that pass through each pair of points and determine whether the lines are parallel or perpendicular. \((2,5)\) and \((5,9)\) \((-1,-1)\) and \((2,3)\)

For the following exercises, use a calculator to help answer the questions. Show that a solution of \(x^{8}-1=0\) is \(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} i\).

Enter the expressions into your graphing utility and find the zeroes to the equation (the \(x\) -intercepts) by using \(2^{\text {nd }}\) CALC 2 :zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero, enter), then right bound (move your cursor to the right of the zero, enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth. $$ \mathrm{Y}_{1}=0.5 x^{2}+x-7 $$

For each of the following exercises, use the graph in the figure below. Find the distance that \((5,2)\) is from the origin. Round to three decimal places.

Solve for \(h: A=\frac{1}{2} b h\)

For the following exercises, write the interval in set-builder notation. $$ (-\infty, 6) $$

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 \(32 \mathrm{ft} /{ }^{2}\) and the period equals \(1 \mathrm{~s}\), fi d the length to the nearest in. (12 in. = \(1 \mathrm{ft}\) ). Round your answer to the nearest in.

Write the interval in set-builder notation. $$ (-\infty, 6) $$

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 Y1, 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, evaluate the expressions, writing the result as a simplified complex number. $$ \frac{1}{i}+\frac{4}{i^{3}} $$

To solve the quadratic equation \(x^{2}+5 x-7=4,\) we can graph these two equations \(\mathrm{Y}_{1}=x^{2}+5 x-7 \qquad \mathrm{Y}_{2}=4\) 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, write the interval in set-builder notation. $$ (4, \infty) $$

Write the interval in set-builder notation. $$ (4, \infty) $$

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 Y1, 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 a model for body surface area, BSA, such that \(B S A=\sqrt{\frac{w h}{3600}},\) where \(w=\) weight in \(\mathrm{kg}\) and \(h=\) height in \(\mathrm{cm} .\) Find the height of a 72 -kg female to the nearest cm whose \(B S A=1.8 .\)

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

To solve the quadratic equation \(0.3 x^{2}+2 x-4=2\) , we can graph these two equations \(\mathrm{Y}_{1}=0.3 x^{2}+2 x-4 \qquad \mathrm{Y}_{2}=2\) and find the points of intersewction. Recall \(2^{\text { nd }} \mathrm{CALC}\) 5 intersection. Do this and find the solutions to the nearest tenth.

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 " \(\mathbf{x}=\) " 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. $$ [-3,5) $$

Write the interval in set-builder notation. $$ [-3,5) $$

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 Y1, 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 a model for body surface area, BSA, such that \(B S A=\sqrt{\frac{w h}{3600}},\) where \(w=\) weight in \(\mathrm{kg}\) and \(h=\) height in \(\mathrm{cm} .\) Find the weight of a 177 -cm male to the nearest kg whose \(B S A=2.1 .\)

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

Solve for \(h : V=\pi r^{2} h\)

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.

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 " \(\mathbf{x}=\) " 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. $$ [-4,1] \cup[9, \infty) $$

Write the interval in set-builder notation. $$ [-4,1] \cup[9, \infty) $$

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

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

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 " \(\mathbf{x}=\) " 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. $$ \mathrm{Y}_{1}=\frac{x+5}{2} $$

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 slope-intercept form.

Solve for \(r : V=\pi r^{2} h\)

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

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.