# MTH Related Rates

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Objectives MTH Related Rates Finding Related Rates We have seen how the Chain Rule can be used to find dy/dx implicitly. Another important use of the Chain Rule is to find the rates of change of two or more related variables that are changing with respect to time. 1

2 Finding Related Rates For example, when water is drained out of a conical tank (see Figure 3.37), the volume V, the radius r, and the height h of the water level are all functions of time t. Volume is related to radius and height. Figure 3.37 Finding Related Rates Knowing that these variables are related by the equation Original equation you can differentiate implicitly with respect to t to obtain the related-rate equation Differentiate with respect to t. From this equation you can see that the rate of change of V is related to the rates of change of both h and r.

3 Example 1 Two Rates That Are Related Suppose x and y are both differentiable functions of t and are related by the equation y = x + 3. Find dy/ when x = 1, given that dx/ = when x = 1. Solution: Using the Chain Rule, you can differentiate both sides of the equation with respect to t. y = x + 3 Write Original equation Differentiate with respect to t. Example 1 Solution cont d Chain Rule When x = 1 and dx/ =, you have 3

4 Example Ripples in a Pond A pebble is dropped into a calm pond, causing ripples in the form of concentric circles, as shown in Figure The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing? Total area increases as the outer radius increases. Figure 3.38 Example Solution The variables r and A are related by A = πr. The rate of change of the radius r is dr/ = 1. Equation: A = πr Given rate: = 1 Find: when r = 4 With this information, you can proceed as in Example 1. 4

5 Example Solution cont d Differentiate with respect to t. Chain Rule Substitute 4 for r and 1 for dr/. When the radius is 4 feet, the area is changing at a rate of 8π square feet per second. Strategy for Solving Related Rate Problems 1. Draw and label a picture. Identify all givenquantities and quantities to be determined. Note what changes and what does not change.. Write down what is given. 3. Write down what is to be found. 4. Write an equation involving the variables whose rates of change either are given or are to be determined. 5. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time. 6. Aftercompleting step 5, substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change. 5

6 Problem Solving with Related Rates The table below lists examples of mathematical models involving rates of change. For instance, the rate of change in the first example is the velocity of a car. EXAMPLE 3: A child throws a stone into a still millpond causing a circular ripple to spread. If the radius increases at the constant rate of 0.5 meter per second, how fast is the area of the ripple increasing when the radius of the ripple is 0 meters? r How fast is the area changing when the radius is 0 meters? Note: rates of change are derivatives. rate of change of radius: rate of change of area: Given: dr/ = 0.5 Find: da/ when r = 0 6

7 We need the formula that connects the radius of the circle with its area. A = πr Next, we differentiate both sides of the equation with respect to t: da = d (πr ) note: r = dr/ da da d =π (r ) dr =π r da =π rr π()(0)(0. 5 ) = 0π m /s or 6.8 m /s EXAMPLE 4: z Two cars start moving from the same point. One travels south at 60mi/h and the other travels west at 5mi/h. At what rate is the distance between the cars increasing two hours later? X Given: dy/ =60 and dx/ = 5 Find: dz/ when t = h Y 7

8 What equation relates the three sides of a right triangle? x + y = z d d d dx dy dz ( x ) + ( y ) = ( z ) x +y = z After hours: x = 50 and y = 10. z = =130 Subst gives: (50)(5) + (10)(60) = (130)(dz/) dz After hours = = (130) 65m/h EXAMPLE 5 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 from the wall? dy/ Given: dx/ = 1 ft/s Find: dy/ when x = 6 ft x 10 ft dx/ = 1 ft/s x + y = 10 d d d x + y = (100) dx dy + y = 0 x eqa 1 8

9 When x = 6: 6 + y = 100 y = = 8 dy Substitute into eqa 1: ()(6)(1) + (8)( ) = 0 16(dy/) = - 1 dy/ = -3/4 ft/s Example 6: The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of cm /min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm Given: dh/ = 1 cm/min da/ = cm /min h = altitude Find: db/ when h = 10cm and A = 100cm A = (1/)bh 100 = (1/)b(10) b = 0 9

10 da d 1 = ( bh) = 1 d (bh) 1 db dh = h+b 1 db = db 1.6cm/min= db = db 8 = 5 dh/ =? when h = 6 ft r Given: dv/ = 9 ft 3 /min Find: dh/ when h = 6 ft 1 V = π r h eqa 1 3 h Ex 7: A water tank in the shape of an inverted circular cone has a base radius of 5 ft and height 10 ft. If water is pumped into the tank at a rate of 9 ft 3 /min, find the rate at which the water level is rising when the water is 6 ft deep. To find a relationship between r and h, use similar triangles: r h h = r =

11 = ( πr h) = π h π h dv d 1 d 1 h d 1 h dv π d 3 dv π dh = ( h ) = 3h 1 1 π dh Substituting gives: 9 = π dh 9(1) dh 9 = = 1 108π dh 1 = 0.3 π ft/min Example: Sand falls from a overhead bin, accumulating in a conical pile with a radius that is always three times its height. If the sand falls from the bin at a rate of 1 ft 3 /min, how fast is the height of the sand pile changing when the pile is 10 ft high? 11

12 Example: A spherical balloon is inflated and its volume increases at a rate of 15 in 3 /min. What is the rate of change of its radius when the radius is 10 in? Example: An inverted conical water tank with a height of 1 ft and a radius of 6 ft is drained through a hole in the vertex at a rate of ft 3 /s. What is the rate of change of the water depth when the water depth is 3 ft? 1

13 An observer stands 00 m from the launch site of a hot-air balloon. The balloon rises vertically at a constant rate of 4 m/s. How fast is the angle of elevation of the balloon increasing 30 s after the launch? As the balloon rises, its distance from the ground y and its angle of elevation θ change simultaneously. θ y 00 m A ladder 5 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of feet per second. (a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 4 feet from the wall? 13

14 (b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall. (c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall. 14

### Example. Suppose you has a 5m ladder resting against a wall. Move the base out at 2 m/s How fast does the top move down the wall? ???

Related rates Example Suppose you has a 5m ladder resting against a wall.??? 5m 2 mps Move the base out at 2 m/s How fast does the top move down the wall? Example Suppose you has a 5m ladder resting against

### Math 226 Quiz IV Review

Math 226 Quiz IV Review. A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3 ft/s. How rapidly is the area enclosed by the ripple increasing at

### respect to x and we did not know what y is. In this case x and y are unknown variables because everything is with respect to t.

.8 Related Rates Section.8 Notes Page This section involves word problems dealing with how something is changing with respect to time. We will first find a formula that is used for the problem and then

### Write one or more equations to express the basic relationships between the variables.

Lecture 13 :Related Rates Please review Lecture 6; Modeling with Equations in our Algebra/Precalculus review on our web page. In this section, we look at situations where two or more variables are related

### Related Rates Problems

These notes closely follow the presentation of the material given in James Stewart s textbook Calculus, Concepts and Contexts (2nd edition). These notes are intended primarily for in-class presentation

### Implicit Differentiation and Related Rates

Implicit Differentiation and Related Rates PART I: Implicit Differentiation Implicit means implied or understood though not directly expressed The equation has an implicit meaning. It implicitly describes

### Miscellaneous Related Rates: Solutions

Miscellaneous Related Rates: Solutions To do these problems, you may need to use one or more of the following: The Pythagorean Theorem, Similar Triangles, Proportionality ( A is proportional to B means

### 1. speed = speed = 7 9 ft/sec. 3. speed = 1 ft/sec. 4. speed = 8 9 ft/sec. 5. speed = 11

Version PREVIEW HW 07 hoffman 57225) This print-out should have 0 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. CalCi02s 00 0.0 points

### Math 220 October 11 I. Exponential growth and decay A. The half-life of fakeium-20 is 40 years. Suppose we have a 100-mg sample.

Math 220 October 11 I. Exponential growth and decay A. The half-life of fakeium-20 is 40 years. Suppose we have a 100-mg sample. 1. Find the mass that remains after t years. 2. How much of the sample remains

### A = πr 2. , the area changes in m2

1 Related Rates So,what sarelatedratewordproblem? Inthese type of word problems two quantities are related somehow, like area and radius of a circle. The area, A, and radius, r, ofacirclearerelatehroughthe

### Math 113 HW #7 Solutions

Math 3 HW #7 Solutions 35 0 Given find /dx by implicit differentiation y 5 + x 2 y 3 = + ye x2 Answer: Differentiating both sides with respect to x yields 5y 4 dx + 2xy3 + x 2 3y 2 ) dx = dx ex2 + y2x)e

### YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

DETAILED SOLUTIONS AND CONCEPTS - SIMPLE GEOMETRIC FIGURES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

### WebAssign Lesson 8-1 Basic Part 2 (Homework)

WebAssign Lesson 8-1 Basic Part 2 (Homework) Current Score : / 24 Due : Thursday, March 27 2014 08:01 AM MDT Shari Dorsey Sp 14 Math 170, section 003, Spring 2014 Instructor: Shari Dorsey 1. /4 points

### Version 005 Exam Review Practice Problems NOT FOR A GRADE alexander (55715) 1. Hence

Version 005 Eam Review Practice Problems NOT FOR A GRADE aleander 5575 This print-out should have 47 questions Multiple-choice questions may continue on the net column or page find all choices before answering

### SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

### Study 2.6, p. 154 # 1 19, Goal: 1. Understand how variables change with respect to time. 2. Understand "with respect to".

Study 2.6, p. 154 # 1 19, 21 25 Goal: 1. Understand how variables change with respect to time. 2. Understand "with respect to". water drop ripples pour cement high rise @ 1:30 water drop ripples pour cement

### 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH. In Isaac Newton's day, one of the biggest problems was poor navigation at sea.

BA01 ENGINEERING MATHEMATICS 01 CHAPTER 3 APPLICATION OF DIFFERENTIATION 3.1 MAXIMUM, MINIMUM AND INFLECTION POINT & SKETCHING THE GRAPH Introduction to Applications of Differentiation In Isaac Newton's

### Contents. 4 Related Rates General Guide for Solving Related Rate Problems Problems Economic Applications...

Contents 4 Related Rates 63 4.1 General Guide for Solving Related Rate Problems........................ 64 4.2 Problems............................................... 64 4.3 Economic Applications........................................

### SURFACE AREA AND VOLUME

SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has

### Basic Math for the Small Public Water Systems Operator

Basic Math for the Small Public Water Systems Operator Small Public Water Systems Technology Assistance Center Penn State Harrisburg Introduction Area In this module we will learn how to calculate the

### a b c d e You have two hours to do this exam. Please write your name on this page, and at the top of page three. GOOD LUCK! 3. a b c d e 12.

MA123 Elem. Calculus Fall 2015 Exam 2 2015-10-22 Name: Sec.: Do not remove this answer page you will turn in the entire exam. No books or notes may be used. You may use an ACT-approved calculator during

### Solution. Section 2.8 Related Rates. Exercise. Exercise. Exercise. dx when y 2. and. 3, then what is when x 1. Solution dy dy.

Section.8 Related Rate Eercie If y and d, then what i when 1 d d 6 6 1 1 6 Eercie If y y and 5, then what i d when y d d y 1 5 5 y 1 d 5 1 55 y Eercie A cube urface area increae at the rate of 7 when the

### Week #15 - Word Problems & Differential Equations Section 8.1

Week #15 - Word Problems & Differential Equations Section 8.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 25 by John Wiley & Sons, Inc. This material is used by

### APPLICATION OF DERIVATIVES

6. Overview 6.. Rate of change of quantities For the function y f (x), d (f (x)) represents the rate of change of y with respect to x. dx Thus if s represents the distance and t the time, then ds represents

### Area of Parallelograms, Triangles, and Trapezoids (pages 314 318)

Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base

### Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

### ACTIVITY: Finding a Formula Experimentally. Work with a partner. Use a paper cup that is shaped like a cone.

8. Volumes of Cones How can you find the volume of a cone? You already know how the volume of a pyramid relates to the volume of a prism. In this activity, you will discover how the volume of a cone relates

### Work. Example. If a block is pushed by a constant force of 200 lb. Through a distance of 20 ft., then the total work done is 4000 ft-lbs. 20 ft.

Work Definition. If a constant force F is exerted on an object, and as a result the object moves a distance d in the direction of the force, then the work done is Fd. Example. If a block is pushed by a

### PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

### Characteristics of the Four Main Geometrical Figures

Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.

### Section 6.1 Angle Measure

Section 6.1 Angle Measure An angle AOB consists of two rays R 1 and R 2 with a common vertex O (see the Figures below. We often interpret an angle as a rotation of the ray R 1 onto R 2. In this case, R

### Answer Key for the Review Packet for Exam #3

Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math Ma-Min Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y

### MCA Formula Review Packet

MCA Formula Review Packet 1 3 4 5 6 7 The MCA-II / BHS Math Plan Page 1 of 15 Copyright 005 by Claude Paradis 8 9 10 1 11 13 14 15 16 17 18 19 0 1 3 4 5 6 7 30 8 9 The MCA-II / BHS Math Plan Page of 15

### HSC Mathematics - Extension 1. Workshop E4

HSC Mathematics - Extension 1 Workshop E4 Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong

### Introduction. Rectangular Volume. Module #5: Geometry-Volume Bus 130 1:15

Module #5: Geometry-olume Bus 0 :5 Introduction In this module, we are going to review the formulas for calculating volumes, rectangular, cylindrical, and irregular shaped. To set the stage, let s suppose

### Unit 9: Areas and volumes of geometrical 3D shapes

Unit 9: Areas and volumes of geometrical 3D shapes In this lesson you will learn about: Pythagoras theorem Areas and volumes of prisms, cubes, pyramids, cylinders, cones ans spheres. Solving problems related

### Section 6.4: Work. We illustrate with an example.

Section 6.4: Work 1. Work Performed by a Constant Force Riemann sums are useful in many aspects of mathematics and the physical sciences than just geometry. To illustrate one of its major uses in physics,

### 0 3 x dx W = 2 x2 = = in-lb x dx W = = 20. = = 160 in-lb.

Friday, October 9, 215 p. 83: 5, 7, 11, 12, 15, 17, 19, 2, 21, 25, 26 Problem 5 Problem. A force of 5 pounds compresses a 15-inch spring a total of 3 inches. How much work is done in compressing the spring

### 4) The length of one diagonal of a rhombus is 12 cm. The measure of the angle opposite that diagonal is 60º. What is the perimeter of the rhombus?

Name Date Period MM2G1. Students will identify and use special right triangles. MM2G1a. Determine the lengths of sides of 30-60 -90 triangles. MM2G1b. Determine the lengths of sides of 45-45 -90 triangles.

### www.mathsbox.org.uk Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx Acceleration Velocity (v) Displacement x

Mechanics 2 : Revision Notes 1. Kinematics and variable acceleration Displacement (x) Velocity (v) Acceleration (a) x = f(t) differentiate v = dx differentiate a = dv = d2 x dt dt dt 2 Acceleration Velocity

### Name: School Team: X = 5 X = 25 X = 40 X = 0.09 X = 15

7th/8th grade Math Meet Name: School Team: Event : Problem Solving (no calculators) Part : Computation ( pts. each) ) / + /x + /0 = X = 5 ) 0% of 5 = x % of X = 5 ) 00 - x = ()()(4) + 6 X = 40 4) 0.6 x

### Section 7.1 Solving Right Triangles

Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,

### Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.

Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles

### Chapter 19. Mensuration of Sphere

8 Chapter 19 19.1 Sphere: A sphere is a solid bounded by a closed surface every point of which is equidistant from a fixed point called the centre. Most familiar examples of a sphere are baseball, tennis

### Solutions to Examples from Related Rates Notes. ds 2 mm/s. da when s 100 mm

Solutions to Examples from Relate Rates Notes 1. A square metal plate is place in a furnace. The quick temperature change causes the metal plate to expan so that its surface area increases an its thickness

### 43 Perimeter and Area

43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study

### Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

### 8-3 Perimeter and Circumference

Learn to find the perimeter of a polygon and the circumference of a circle. 8-3 Perimeter Insert Lesson and Title Circumference Here perimeter circumference Vocabulary The distance around a geometric figure

### Active Calculus & Mathematical Modeling Activities and Voting Questions Carroll College MA 122. Carroll College Mathematics Department

Active Calculus & Mathematical Modeling Activities and Voting Questions Carroll College MA 122 Carroll College Mathematics Department Last Update: June 2, 2015 2 To The Student This packet is NOT your

### Basic Lesson: Pythagorean Theorem

Basic Lesson: Pythagorean Theorem Basic skill One leg of a triangle is 10 cm and other leg is of 24 cm. Find out the hypotenuse? Here we have AB = 10 and BC = 24 Using the Pythagorean Theorem AC 2 = AB

### Unit 1 - Radian and Degree Measure Classwork

Unit - Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side - starting and ending Position of the ray Standard position origin if the vertex,

### Calculus 1st Semester Final Review

Calculus st Semester Final Review Use the graph to find lim f ( ) (if it eists) 0 9 Determine the value of c so that f() is continuous on the entire real line if f ( ) R S T, c /, > 0 Find the limit: lim

To find a maximum or minimum: Find an expression for the quantity you are trying to maximise/minimise (y say) in terms of one other variable (x). dy Find an expression for and put it equal to 0. Solve

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### SURFACE AREAS AND VOLUMES

CHAPTER 1 SURFACE AREAS AND VOLUMES (A) Main Concepts and Results Cuboid whose length l, breadth b and height h (a) Volume of cuboid lbh (b) Total surface area of cuboid 2 ( lb + bh + hl ) (c) Lateral

### Area is a measure of how much space is occupied by a figure. 1cm 1cm

Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number

### Projectiles Problem Solving Constant velocity: x=d=vt

Name: Date: Projectiles Problem Solving Constant velocity: x=d=vt Acceleration: x=d= a 2 t 2 +v 0 t ; v=at+v 0 ; x=d= v 0 +v 2 t ; v2 =v 0 2 +2 ad 1. Given the following situation of a marble in motion

### Pythagorean Theorem, Distance and Midpoints Chapter Questions. 3. What types of lines do we need to use the distance and midpoint formulas for?

Pythagorean Theorem, Distance and Midpoints Chapter Questions 1. How is the formula for the Pythagorean Theorem derived? 2. What type of triangle uses the Pythagorean Theorem? 3. What types of lines do

### Solids. Objective A: Volume of a Solids

Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular

### Geometry Work Samples -Practice-

1. If ABCD is a square and ABE is an equilateral triangle, then what is the measure of angle AFE? (H.1G.5) 2. Square ABCD has a circular arc of radius 2 with the center at A and another circular arc with

### Geometry Chapter 12. Volume. Surface Area. Similar shapes ratio area & volume

Geometry Chapter 12 Volume Surface Area Similar shapes ratio area & volume Date Due Section Topics Assignment Written Exercises 12.1 Prisms Altitude Lateral Faces/Edges Right vs. Oblique Cylinders 12.3

### WebAssign Lesson 3-3 Applications (Homework)

WebAssign Lesson 3-3 Applications (Homework) Current Score : / 27 Due : Tuesday, July 15 2014 11:00 AM MDT Jaimos Skriletz Math 175, section 31, Summer 2 2014 Instructor: Jaimos Skriletz 1. /3 points A

### Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)

Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.

### Geometry Review. Here are some formulas and concepts that you will need to review before working on the practice exam.

Geometry Review Here are some formulas and concepts that you will need to review before working on the practice eam. Triangles o Perimeter or the distance around the triangle is found by adding all of

### Unit 1 - Radian and Degree Measure Classwork

Unit 1 - Radian and Degree Measure Classwork Definitions to know: Trigonometry triangle measurement Initial side, terminal side - starting and ending Position of the ray Standard position origin if the

### Solutions to Homework 10

Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x

### Height. Right Prism. Dates, assignments, and quizzes subject to change without advance notice.

Name: Period GL UNIT 11: SOLIDS I can define, identify and illustrate the following terms: Face Isometric View Net Edge Polyhedron Volume Vertex Cylinder Hemisphere Cone Cross section Height Pyramid Prism

### WEIGHTS AND MEASURES. Linear Measure. 1 Foot12 inches. 1 Yard 3 feet - 36 inches. 1 Rod 5 1/2 yards - 16 1/2 feet

WEIGHTS AND MEASURES Linear Measure 1 Foot12 inches 1 Yard 3 feet - 36 inches 1 Rod 5 1/2 yards - 16 1/2 feet 1 Furlong 40 rods - 220 yards - 660 feet 1 Mile 8 furlongs - 320 rods - 1,760 yards 5,280 feet

### Northfield Mount Hermon School Advanced Placement Calculus. Problems in the Use of the Definite Integral

Northfield Mount Hermon School Advanced Placement Calculus Problems in the Use of the Definite Integral Dick Peller Winter 999 PROBLEM. Oil is leaking from a tanker at the rate of R(t) = e -.t gallons

### Maximum / Minimum Problems

171 CHAPTER 6 Maximum / Minimum Problems Methods for solving practical maximum or minimum problems will be examined by examples. Example Question: The material for the square base of a rectangular box

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Platonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples:

Solid Geometry Solid Geometry is the geometry of three-dimensional space, the kind of space we live in. Three Dimensions It is called three-dimensional or 3D because there are three dimensions: width,

### Homework #1 Solutions

MAT 303 Spring 203 Homework # Solutions Problems Section.:, 4, 6, 34, 40 Section.2:, 4, 8, 30, 42 Section.4:, 2, 3, 4, 8, 22, 24, 46... Verify that y = x 3 + 7 is a solution to y = 3x 2. Solution: From

### Lesson 11: The Volume Formula of a Pyramid and Cone

Classwork Exploratory Challenge Use the provided manipulatives to aid you in answering the questions below. a. i. What is the formula to find the area of a triangle? ii. Explain why the formula works.

### Math 32B Practice Midterm 1

Math B Practice Midterm. Let be the region bounded b the curves { ( e)x + and e x. Note that these curves intersect at the points (, ) and (, e) and that e

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### Area Long-Term Memory Review Review 1

Review 1 1. To find the perimeter of any shape you all sides of the shape.. To find the area of a square, you the length and width. 4. What best identifies the following shape. Find the area and perimeter

### Finding Volume of Rectangular Prisms

MA.FL.7.G.2.1 Justify and apply formulas for surface area and volume of pyramids, prisms, cylinders, and cones. MA.7.G.2.2 Use formulas to find surface areas and volume of three-dimensional composite shapes.

### MENSURATION. Definition

MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters

### Davis Buenger Math Solutions September 8, 2015

Davis Buenger Math 117 6.7 Solutions September 8, 15 1. Find the mass{ of the thin bar with density 1 x function ρ(x) 1 + x < x. Solution: As indicated by the box above, to find the mass of a linear object

### MA123 Elem. Calculus Fall 2013 Exam

MA123 Elem Calculus Fall 2013 Exam 3 2013-11-21 Name: Sec: Do not remove this answer page you will turn in the entire exam You have two hours to do this exam No books or notes may be used You may use a

### Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =

Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a

### Perimeter is the length of the boundary of a two dimensional figure.

Section 2.2: Perimeter and Area Perimeter is the length of the boundary of a two dimensional figure. The perimeter of a circle is called the circumference. The perimeter of any two dimensional figure whose

### Calculus (6th edition) by James Stewart

Calculus (6th edition) by James Stewart Section 3.8- Related Rates 9. If and find when and Differentiate both sides with respect to. Remember that, and similarly and So we get Solve for The only thing

### The small increase in x is. and the corresponding increase in y is. Therefore

Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that

### Section 7.2 Area. The Area of Rectangles and Triangles

Section 7. Area The Area of Rectangles and Triangles We encounter two dimensional objects all the time. We see objects that take on the shapes similar to squares, rectangle, trapezoids, triangles, and

### Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

### To find a maximum or minimum: Find an expression for the quantity you are trying to maximise/minimise (y, say) in terms of one other variable (x).

Maxima and minima In this activity you will learn how to use differentiation to find maximum and minimum values of functions. You will then put this into practice on functions that model practical contexts.

### Exam 1 Review Questions PHY 2425 - Exam 1

Exam 1 Review Questions PHY 2425 - Exam 1 Exam 1H Rev Ques.doc - 1 - Section: 1 7 Topic: General Properties of Vectors Type: Conceptual 1 Given vector A, the vector 3 A A) has a magnitude 3 times that

### Education Resources. This section is designed to provide examples which develop routine skills necessary for completion of this section.

Education Resources Circle Higher Mathematics Supplementary Resources Section A This section is designed to provide examples which develop routine skills necessary for completion of this section. R1 I

### Perimeter, Area, and Volume

Perimeter, Area, and Volume Perimeter of Common Geometric Figures The perimeter of a geometric figure is defined as the distance around the outside of the figure. Perimeter is calculated by adding all

### Area of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in

Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 Solve: Find the area of each triangle. 1. 2. 3. 5in4in 11in 12in 9in 21in 14in 19in 13in

### PROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS

PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,

### 1.7 Find Perimeter, Circumference,

.7 Find Perimeter, Circumference, and rea Goal p Find dimensions of polygons. Your Notes FORMULS FOR PERIMETER P, RE, ND CIRCUMFERENCE C Square Rectangle side length s length l and width w P 5 P 5 s 5

Section.1 Radian Measure Another way of measuring angles is with radians. This allows us to write the trigonometric functions as functions of a real number, not just degrees. A central angle is an angle

### PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

### MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010

MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic