Use our 12th grade math worksheets. rev2023.4.21.43403. Our mission is to transform the way children learn math, to help them excel in school and competitive exams. Example 3: Solve for h. 5i= (6hi+2)(5i8) 5 i = ( 6 h i + 2) ( 5 i 8) 1) This is a complex math equation that involves complex numbers and an imaginary . Question 1 For the function f (x) = (x + 3) (x - 2) 3 find Help. For a polynomial $p(x)$, the value of $p(3)$ is $-2$. For example, if you want the equation for sample standard deviation you could navigate to the relevant Wikipedia article (here), go down to the section that has the relevant equation, then click on "edit" to see the LaTeX code for the equation. The circumference, $C$, of a circle is $C = 2r$, where $r$ is the radius of the circle. If 13 x 12 = 651 & 41 x 23 = 448, then, 24 x 22 =? Your text : Keyboard Layout : Arithmetic & Algebra Superscript & Subscript Fractions Statistics Measurements Calculus Greek symbols Letters symbols Logic & Theory Geometry Since triangle DEF is similar to triangle ABC, with vertex F corresponding to vertex C, the measure of $\angle {F}$ equals the measure of $\angle {C}$. Our guide will help you beat the clock and maximize your score. Embedded hyperlinks in a thesis or research paper, How to convert a sequence of integers into a monomial, Generating points along line with specifying the origin of point generation in QGIS. Let $x$ be the number of left-handed female students and let $y$ be the number of left-handed male students. How to get a number 100 by using four sevens (7s) and a one (1)? Thus, by Little's law, there are, on average, $N = rT = (1.5)(12) = 18$ shoppers in the new store at any time. Replace the question mark in the above problem with the appropriate number. For example, if the answer is 42.1%, enter 42.1), ANSWER EXPLANATION: According to the original information given, the estimated average number of shoppers in the original store at any time (N) is 45. Why did US v. Assange skip the court of appeal? For the new store, the owner estimates that, during business hours, an average of 90 shoppers per hour enter the store and each of them stays an average of 12 minutes. This calculator follows standard rules to solve equations. This equals, $$({8-i}/{3-2i})({3+2i}/{3+2i})={24+16i-3+(-i)(2i)}/{(3^2)-(2i)^2}$$, Since $i^2=-1$, this last fraction can be reduced simplified to, $$ {24+16i-3i+2}/{9-(-4)}={26+13i}/{13}$$. This is true only one day in a year. No Calculator SAT Math Questions Question 1 C = 5 9 ( F 32) The equation above shows how temperature F, measured in degrees Fahrenheit, relates to a temperature C, measured in degrees Celsius. For using LaTeX with WordPress you might be interested in. This is. 4 children get 1 apple each while the fifth child gets the basket with the remaining apple still in it. Once you are done, you can save to file or copy [Ctrl]+ [c] & paste [Ctrl]+ [v] it to other documents or to your email. It's called a Diophantine Equation, and it's sometimes known as the "summing of three cubes": Find x, y, and z such that x+y+z=k, for each k from one to 100. There are two cut and paste activity pages for the associative property. The question states that $p(3) = -2$, so it must be true that. Just type in the math problem into the interactive calculator using its prescribed syntax, and it will give you the correct answer. x^Zms_~'% HdvMwN"6'HTg Rv&i>p\%y$|xvU$TervUe*9H^=g0&YQ)j'4r/Tf6VLgS*Mg?HY-[~F#p)H._c6 eIg{"m2FDeV&Y !yR*I6Mry k-5^D&?i*|ps*Yqu`VEkTj61n%9s|-w[nwfGf]/?6. Open the document where you want to paste the copied equation. Can I connect multiple USB 2.0 females to a MEAN WELL 5V 10A power supply? What day is my Birthday? Then, among Rs.5, Each person has taken Rs.1 and remaining Rs.2 given to the beggar beside the shop. I am an odd number. ANSWER EXPLANATION: If the polynomial $p(x)$ is divided by a polynomial of the form $x+k$ (which accounts for all of the possible answer choices in this question), the result can be written as. In these acronyms, "brackets" are the same as parentheses, and "order" is the same as exponents. In the question, it states that, in the new store, the manager estimates that an average of 90 shoppers per hour (60 minutes) enter the store, which is equivalent to 1.5 shoppers per minute (r). Look at this series: 36, 34, 30, 28, 24, What number should come next? Our expert tutors conduct 2 or more live classes per week, at a pace that matches the child's learning needs. K5 Technology Lab. The consent submitted will only be used for data processing originating from this website. The dynamic pressure $q$ generated by a fluid moving with velocity $v$ can be found using the formula above, where $n$ is the constant density of the fluid. The store owner determines that, during business hours, approximately 84 shoppers per hour make a purchase and each of these shoppers spend an average of 5 minutes in the checkout line. Copy and paste R commands and outputs. Pandemic Pods, Microschools & Homeschool Co-ops: An Overview. percent less than the average number of shoppers in the original store at any time. << /Names 70 0 R /OpenAction 80 0 R /Outlines 63 0 R /PageMode /UseOutlines /Pages 62 0 R /Type /Catalog >> The first letter of each word in the phrase creates the PEMDAS acronym. 41 years ago, when Sally was 13 and her mother was 39. PEMDAS is typcially expanded into the phrase, "Please Excuse My Dear Aunt Sally." The golden rule is to know that maths is a mindful activity rather than a task. derivative to determine the intervals of increase and decrease of a function, the maximum and minimum points, the interval(s) of concavity and points of inflections are discussed. ANSWER EXPLANATION: The equation $f(x) = k$ gives the solutions to the system of equations. Thats it. For example, 5 - 3 + 2 = 4 and 5 - 3 + 2 does not equal 0. Mathway can be that extra eyeto help double-check your work. WARNING: Since there are a limited number of official SAT practice tests, you may want to wait to read this article until you've attempted all or most of the first four official practice tests (since most of the questions below were taken from those tests). Acoustic plug-in not working at home but works at Guitar Center. Using only an addition, how do you add eight 8s and get the number 1000? We'll also give you a step-by-step, custom program to follow so you'll never be confused about what to study next. (Note: Assume that none of the eighth-grade students are both right-handed and left-handed.). Want to know what makes these questions so difficult and how best to solve them? If $x$ is the average (arithmetic mean) of $m$ and $9$, $y$ is the average of $2m$ and $15$, and $z$ is the average of $3m$ and $18$, what is the average of $x$, $y$, and $z$ in terms of $m$? in Physics and Engineering, Exercises de Mathematiques Utilisant les Applets, Trigonometry Tutorials and Problems for Self Tests, Elementary Statistics and Probability Tutorials and Problems, Free Practice for SAT, ACT and Compass Math tests. A Comprehensive Guide. Two girls were born to the same mother, at the same time, on the same day, in the same month and the same year and yet somehow theyre not twins. Little's law can be applied to any part of the store, such as a particular department or the checkout lines. If you don't know your way around functions, this would be a tricky problem. Of the following, which is closest to the volume of the grain silo, in cubic feet? By squaring $1.5$, you can rewrite the previous equation as, $$q_2 = (2.25)({1}/{2})n(v_1)^2 = (2.25)q_1$$, Therefore, the ratio of the dynamic pressure of the faster fluid is. Aiming for a perfect score? Thus, 5*10, or 50, of the 122 right-handed students are female. Math Worksheets. +Addition Why not? The uses of the first and second What number am I? Once you've got a good idea of your current level and percentile ranking, you can set milestones and goals for your ultimate SAT Math score. Believe it or not, this math question actually requires no math whatsoever. Uh oh! Want to improve your SAT score by 160 points? You can hold [Shift] for the upper case Greek characters. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Note that some sections will have more problems than others and some will have more or less of a variety of problems. You should then multiply $(-8x-3)$ and $(ax-2)$ using FOIL. Understand the Cuemath Fee structure and sign up for a free trial. ANSWER EXPLANATION: To figure out the answer to this question, you'll first need to know the formula for finding the circumference of a circle. Khan Academy's 100,000+ free practice questions give instant feedback, don't need to be graded, and don't require a printer. It's important to understand what makes these hard questions "hard." You can also include parentheses and numbers with exponents or roots in your equations. This equation can be written as "|x + 1 - x^2| = 2 (2x^2 - 1)." To solve this equation, you can follow these steps: Move all the terms to one side of the equation, so that the left-hand side is equal to zero. Multiplication and Division are performed as they occur in the equation, from left to right. The second digit is four times as big as the third digit, while the first digit is three less than the second digit. We can use that as a good starting point, and break up the problem in pieces using this term as the basic building block: fp(x) = 9 (x2 + p) 1 x2 + 2. If you would like to change your settings or withdraw consent at any time, the link to do so is in our privacy policy accessible from our home page.. According to the Pythagorean theorem, $$AB ={20^2-16^2}={400-256}={144}=12$$. Treating maths as a language totally eradicates the concept of hard math problems or tricky maths questions from your mind. Unlimited AI-generated practice problems and answers. The day before yesterday I was 25. Answer: 9. If $p0$, what is the value of $r/p$? $$24x^2 + 25x - 47 = -8ax^2 - 3ax +16x + 6 - 53$$, Then, reduce on the right side of the equation, $$24x^2 + 25x - 47 = -8ax^2 - 3ax +16x - 47$$. A way to remember this could be to write PEMDAS as PE(MD)(AS) or BEDMAS as BE(DM)(AS). For example, in the equation 4 divided by you must enter it as 4/(1/2). Maths is nothing less than a game, a game that polishes your intelligence and boosts your concentration. Since the two cones have identical dimensions, the total volume, in cubic feet, of the silo is given by, $$V_{silo}=(5^2)(10)+(2)({1}/{3})(5^2)(5)=({4}/{3})(250)$$. The owner of the Good Deals Store opens a new store across town. However, there are times when a teacher or tutor is not available to help you solve a math problem. From the side lengths of triangle ABC, $$sinF ={\opposite \side}/{\hypotenuse}={AB}/{AC}={12}/{20}={3}/{5}$$, The incomplete table above summarizes the number of left-handed students and right-handed students by gender for the eighth grade students at Keisel Middle School. If you already know how the area of a parallelogram and the area of a triangle are related, then adding 79 and 10 and subsequently subtracting 72 and 8 to get 9 should make sense. Note that while this is in the calculator section of the SAT, you absolutely do not need your calculator to solve it! To view the purposes they believe they have legitimate interest for, or to object to this data processing use the vendor list link below. All rights reserved. You can also include parentheses and numbers with exponents or roots in your equations. Want to improve your SAT score by 160 points? Now pour 2 litres into the 5Lt bottle until it becomes full. Look at this series: 22, 21, 23, 22, 24, 23, What number should come next? which is approximately equal to 1,047.2 cubic feet. The uses of the first and second derivative to determine the intervals of increase and decrease of a function, the maximum and minimum points, the interval (s) of concavity and points of inflections are discussed. This problem comes straight from a standardized test given in New York in 2014. We and our partners use data for Personalised ads and content, ad and content measurement, audience insights and product development. Don't worry too much about the no-calculator section, though: if you're not allowed to use a calculator on a question, it means you don't need a calculator to answer it. If you would like to change your settings or withdraw consent at any time, the link to do so is in our privacy policy accessible from our home page.. You appear to be on a device with a "narrow" screen width (, 2.4 Equations With More Than One Variable, 2.9 Equations Reducible to Quadratic in Form, 4.1 Lines, Circles and Piecewise Functions, 1.5 Trig Equations with Calculators, Part I, 1.6 Trig Equations with Calculators, Part II, 3.6 Derivatives of Exponential and Logarithm Functions, 3.7 Derivatives of Inverse Trig Functions, 4.10 L'Hospital's Rule and Indeterminate Forms, 5.3 Substitution Rule for Indefinite Integrals, 5.8 Substitution Rule for Definite Integrals, 6.3 Volumes of Solids of Revolution / Method of Rings, 6.4 Volumes of Solids of Revolution/Method of Cylinders, A.2 Proof of Various Derivative Properties, A.4 Proofs of Derivative Applications Facts, 7.9 Comparison Test for Improper Integrals, 9. 'https:':'http:')+'//cse.google.com/cse.js?cx='+cx;var s=document.getElementsByTagName('script')[0];s.parentNode.insertBefore(gcse,s)}, You may solve a set of 10 questions with their detailed solutions and also a set of 50 questions, with their answers, in the applet to self test you background on how to. Math Equation Solver | Order of Operations, use numbers and + - * / ^ r . Click on the " Solution " link for each problem to go to the page containing the solution. For the given circle with a radius of 1, the circumference is $C = 2()(1)$, or $C = 2$. For example: If you want an entry such as 1/2 to be treated as a fraction then enter it as (1/2). What is the number of parking space covered by the car? High school is a crucial junction in a student's educational journey. Cite this content, page or calculator as: Furey, Edward "Math Equation Solver | Order of Operations" at https://www.calculatorsoup.com/calculators/math/math-equation-solver.php from CalculatorSoup, To solve this fun maths question, you need to understand how the area of a parallelogram works. fivem gun crafting location; pros and cons of lifesource water system; usta friend at court 2022 handbook What is the value of $sinF$? total amount. Question 1 Simplify: 9 log9(4) = A. Secret to success: Take your time, analyze what is being asked of you, and draw a diagram if it's helpful to you. you don't need to "edit" to access the wikipedia TeX code, just rigth-click (or equivalent) in the equation image and look for "image info" (or equivalent) and you can access the field "associated text" which is the raw TeX code that you can copy. Looking for job perks? A) I onlyB) II onlyC) III onlyD) I and II only, ANSWER EXPLANATION: Think of the equation as an equation for a line. Printable Counting Worksheets You'll find many pages here: counting; skip counting; counting by halves; counting to one hundred; hundreds charts; manipulatives; 12 Days of Christmas. << /Filter /FlateDecode /Length 3304 >> Since ${9}/{5}$ = 1.8, statement II is true. Boundary Value Problems & Fourier Series, 8.3 Periodic Functions & Orthogonal Functions, 9.6 Heat Equation with Non-Zero Temperature Boundaries, 1.14 Absolute Value Equations and Inequalities, Solving Trig Equations with Calculators, Part I, Solving Trig Equations with Calculators, Part II, Derivatives of Exponential and Logarithm Functions, LHospitals Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution / Method of Cylinders. This division can be represented by $/3 * {1/2} = 1/6$. We and our partners use cookies to Store and/or access information on a device. Each output is a random math problem. Press Alt with the appropriate letter. So, the total effective amount paid is 9*3 = 27 and the amount given to beggar is Rs.2, thus the total is Rs.29. The most amazing part of Wolfram Problem Generator is something you can't even see. ANSWER EXPLANATION: There are two ways to solve this question. Without further ado, they are: X = -80538738812075974, Y = 80435758145817515, and Z = 12602123297335631. Calculus questions with detailed solutions are presented. If you flip the image upside down, youll see that what youre dealing with is a simple number sequence. In triangle $ABC$, the measure of $B$ is 90, $BC=16$, and $AC$=20. Some examples. Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor: Step 2 : Let Algebra Helper solve it: Step 3 : Ask for an explanation for the steps you don't understand: Algebra Helper can solve problems in all the following areas: simplification of algebraic expressions (operations with polynomials . Most sections should have a range of difficulty levels in . Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. This seems more managable to me. To be honest, I start wondering what this math solver cannot do! Therefore, the average number of shoppers, $N$, in the checkout line at any time during business hours is 7. Since $x + k$ is a degree-1 polynomial (meaning it only includes $x^1$ and no higher exponents), the remainder is a real number. Is there a weapon that has the heavy property and the finesse property (or could this be obtained)? Things to Consider Homeschooling a Highschool Student. Therefore, statement I is true. With all of these steps, it would probably be faster to just write the LaTeX equations by hand; this depends of course on the specific equation. Using the information given in the problem, the number of right-handed female students will be $5x$ and the number of right-handed male students will be $9y$. Pro subscribers can also create printable worksheets for study sessions and quizzes. hbspt.cta.load(360031, '4efd5fbd-40d7-4b12-8674-6c4f312edd05', {}); Have any questions about this article or other topics? Sample 1 College algebra multiple choice questions, with answers, are presented. A real solution of a system of two equations corresponds to a point of intersection of the graphs of the two equations in the $xy$-plane. If you don't know how, you can find instructions. Moreover, you can also use this as a graphing utility to graph any functions such as linear, quadratic, trigonometric functions, etc. Ask questions; get answers. Division and Multiplication, Addition Learn more about Stack Overflow the company, and our products. Since equations are in math mode, you will need to paste in between $ signs. What is the ratio of the dynamic pressure of the faster fluid to the dynamic pressure of the slower fluid? Cuemath is one of the world's leading math learning platforms that offers LIVE 1-to-1 online math classes for grades K-12. A ship anchored in a port has a ladder which hangs over the side. This includes the inception of the ideology that maths is simpler than their fear. Take away one letter and I become even. In this case, the answer is 4 x (2 x 3). Again fill the 5Lt bottle and pour 1 litre into 3 Lt bottle until it becomes full. I'd also just love a bunch of practice problems I could copy the original LaTeX code and paste right into my text editor. This can be frustrating at times. ANSWER EXPLANATION: The volume of the grain silo can be found by adding the volumes of all the solids of which it is composed (a cylinder and two cones). For example, 4/2*2 = 4 and 4/2*2 does not equal 1. $-2 = p(3) = (3 + (-2))q(3) + 0$$-2 = (3-2)q(3)$$-2 = (-1)q(3)$, C. $-2 = p(3) = (3 + 2)q(3) + 0$$-2 = (5)q(3)$, This could be true, but only if $q(3)={-2}/{5}$, D. $-2 = p(3) = (3 + (-3))q(3) + (-2)$$-2 = (3 - 3)q(3) + (-2)$$-2 = (0)q(3) + (-2)$. So, they paid Rs.30 i.e. BEDMAS stands for "Brackets, Exponents, The length of the ladder is 200cm, the distance between each rung in 20cm and the bottom rung touches the water. Problem 5: Find the x intercept of the graph of the equation . $-2 = p(3) = (3 + (-5))q(3) + 0$$-2=(3-5)q(3)$$-2=(-2)q(3)$, B. To view the purposes they believe they have legitimate interest for, or to object to this data processing use the vendor list link below. Shift + click a button to insert its upper-case form. We've curated 15 of the most difficult SAT Math questions for you to try below, along with walkthroughs of how to get the answer (if you're stumped). Click "Generate" several times for a different output. . Answer: is 3, because six has three letters. With the right attitude and friends and teachers, doing math can be most entertaining and delightful. Some of our partners may process your data as a part of their legitimate business interest without asking for consent. Practising maths with the aid of dice, cards, puzzles and tables reassures that your child effectively approaches Maths. Solve math problems using order of operations like PEMDAS, BEDMAS, BODMAS, GEMDAS and MDAS. Substituting the expressions in m for each variable ($x$, $y$, $z$) gives, $$[{m+9}/{2}+{2m+15}/{2}+{3m+18}/{2}]/3$$. I prepare lessons for my students in printed form as well as on a Wordpress blog. If $3x-y = 12$, what is the value of ${8^x}/{2^y}$? hbspt.cta._relativeUrls=true;hbspt.cta.load(360031, '2bf868a5-3c9b-4b35-bf8a-78b84d7c255a', {"useNewLoader":"true","region":"na1"}); Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. The fraction $1/6$ can also be rewritten as $0.166$ or $0.167$. The logic is payments should be equal to receipts. Possible answers are provided in a number bank. If you're aiming for perfect (or close to), then you'll need to know what the most difficult SAT math questions look like and how to solve them. If you're worried about spoiling those tests, stop reading this guide now; come back and read it when you've completed them. The same confusion can also happen with "AS" however, addition and subtraction also have the same precedence and are performed during the same step from left to right. Solving Complex Equation Examples. A temperature increase of 1 degree Fahrenheit is equivalent to a temperature increase of $5/9$ degree Celsius. No need to draw conclusions. If you liked this article, you'll love our classes. Cuemath, a student-friendly mathematics and coding platform, conducts regular OnlineClasses for academics and skill-development, and their Mental Math App, on both iOS and Android, is a one-stop solution for kids to develop multiple skills. At any time during business hours, about how many shoppers, on average, are waiting in the checkout line to make a purchase at the Good Deals Store? This tricky math problem went viral a few years back after it appeared on an entrance exam in Hong Kong for six-year-olds. Pour remaining 2 litres in 5Lt bottle into 3Lt bottle. by Guest Tue Jan 03, 2023 1:58 pm. What does "up to" mean in "is first up to launch"? Math worksheets take forever to hunt down across the internet. << /Linearized 1 /L 215330 /H [ 1592 289 ] /O 30 /E 161585 /N 12 /T 214905 >> Where has the other Rs.1 gone from the original Rs.30? But, dont worry. Generate printable worksheets. To find what fraction of the circumference the length of ${AB}$ is, divide the length of the arc by the circumference, which gives $/3 2$. Check out our best-in-class online SAT prep classes. Why does contour plot not show point(s) where function has a discontinuity? However, the ascending difficulty resets from easy to hard on the grid-ins. She has years of tutoring experience and writes creative works in her free time. Use the following information for both question 7 and question 8. Then how many days does it take him to reach the mountain top? See how other students and parents are navigating high school, college, and the college admissions process. Of course, its not actually possible for half a dog to compete in a dog show, but for the sake of this math problem lets assume that it is. 1. Springer offers a very nice utility to search for equations at http://latexsearch.com/. Therefore, $sin F = sin C$. However, multiplication and division have the same precedence. A man is climbing up a mountain which is inclined. Parametric Equations and Polar Coordinates, 9.5 Surface Area with Parametric Equations, 9.11 Arc Length and Surface Area Revisited, 10.7 Comparison Test/Limit Comparison Test, 12.8 Tangent, Normal and Binormal Vectors, 13.3 Interpretations of Partial Derivatives, 14.1 Tangent Planes and Linear Approximations, 14.2 Gradient Vector, Tangent Planes and Normal Lines, 15.3 Double Integrals over General Regions, 15.4 Double Integrals in Polar Coordinates, 15.6 Triple Integrals in Cylindrical Coordinates, 15.7 Triple Integrals in Spherical Coordinates, 16.5 Fundamental Theorem for Line Integrals, 3.8 Nonhomogeneous Differential Equations, 4.5 Solving IVP's with Laplace Transforms, 7.2 Linear Homogeneous Differential Equations, 8. Each of the 10 multiplication activities and 10 division activities in this FREE Math Cut & Paste - Multiplication and Division unit includes 8 math problems. To solve this fun maths question, you need to understand how the area of a parallelogram works. This question comes directly from a second grader's math homework. Alt + click a button to copy a single character to the clipboard.. You can select text and press Ctrl + C to copy it to your document. Exhausted, he then takes rest there at night time. Knowing how to handle the hardest questions the test can throw at you will make taking the real SAT seem a lot less daunting. A grain silo is built from two right circular cones and a right circular cylinder with internal measurements represented by the figure above. Algebra Problems with Detailed Solutions Problem 1: Solve the equation 5 (- 3x - 2) - (x - 3) = -4 (4x + 5) + 13 Detailed Solution. Now that you're sure you should be attempting these questions, let's dive right in! The average number of shoppers in the new store at any time is what percent less than the average number of shoppers in the original store at any time? Dividing a positive by a negative or a negative by a positive produces a negative result. Therefore, $\ov {AC}$ is the hypotenuse of right triangle ABC, and $\ov {AB}$ and $\ov {BC}$ are the legs of right triangle ABC. The AP Calculus AB exam will be offered both on paper and digitally in 2021. If signs are the same then keep the sign and add the numbers. Students may also use the keyboard shortcuts "Ctrl-C" (copy) and "Ctrl-V" (paste). Continue with Recommended Cookies, var loadCseCallback=function(){var r=document.querySelector('.gsc-placeholder-table');r.parentNode.removeChild(r);document.getElementById("gsc-i-id1").focus()};window.__gcse={callback:loadCseCallback};function loadCSE(i){var cx='002033744443348646021:uhlxwcaqasa';var gcse=document.createElement('script');gcse.type='text/javascript';gcse.async=true;gcse.src=(document.location.protocol=='https:'? Here is a listing of sections for which practice problems have been written as well as a brief description of the material covered in the notes for that particular section. Since there are 60 minutes in one hour, the rate is ${84 \shoppers \per \hour}/{60 \minutes} = 1.4$ shoppers per minute. I note the " x2 + something " term in both the numerator and denominator.
calculus problems copy and paste