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MATH 1300 Lecture Notes - Lecture 29: Pythagorean Theorem

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10 Nov 2018

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York University

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Mathematics and Statistics

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MATH 1300 verified notes

MATH 1300 Lecture Notes - Lecture 27: Compound Interest

MATH 1300 Lecture Notes - Lecture 29: Pythagorean Theorem

MATH 1300 Lecture Notes - Lecture 30: Pythagorean Theorem, Searchlight, Differentiable Function

Document Summary

Math 1300 lecture 29 notes related rates. In order to connect dvydt and drydt, we first relate v and r by the formula for the volume of a sphere: In order to use the given information, we differentiate each side of this equation with respect to t: to differentiate the right side, we need to use the chain rule, now we solve for the unknown quantity: In this problem, the relationship between x and y is given by the. It is useful to recall some of the problem-solving principles and adapt them to related rates in light of our experience: read the problem carefully, draw a diagram if possible. Introduce notation: assign symbols to all quantities that are functions of time, express the given information and the required rate in terms of derivatives, write an equation that relates the various quantities of the problem.

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Related Questions

To solve a related rates problem, follow the steps: 1. Sketch a diagram if possible. 2. Write down all the variables along with the rates given. 3. Write down the equation that relates all the variables. 4. Differentiate the equation implicitly. 5. Solve for the unknown rate.

1. Suppose a spherical balloon is inflated at the rate of 10 cubic centimeters per minute. How fast is the radius of the balloon increasing at the time when the radius is 5 cm? Recall that the formula for the volume of a sphere is V = 4/3 Ïr3.

2. A 20-foot ladder is leaning against the wall. If the base of the ladder is sliding away from the wall at the rate of 3 feet per second, find the rate at which the top of the ladder is sliding down when the top of the ladder is 8 feet from the ground.

3. Water leaking onto a floor creates a circular puddle with an area that increases at the rate of 3 square centimeters per minute. How fast is the radius of the puddle increasing when the radius is 10 cm? Recall that the formula for the area of a circle is A = r2Ï.

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Ed 9th B Blackboarc x h, ed rds.pd Adobe Read Wind 178 1334 100% Tools Sign (b) If 0 is increasing at the rate of radian per minute, find the when rates of change of the area when 0 /6 and 6 m/3 Explain why the rate of change of the area of the triangle is when not constant even though d6/dt is constant. 16. Volume The radius of a sphere is increasing at a rate of when 3 inches per minute a) Find the rates of c hange of the volume when 9 inches when and 36 inches b) Explain why the rate of change of the volume of the sphere when is not constant even though dr/dt is constant. 17. Volume A spherical balloon is inflated with gas at the rate of when 3, y 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is a) 30 centi and (b) 60 centimeters when 4, y 18. Volume All edges of a cube are expanding at a ate of 6 centimeters per second. How fast is e volume c hanging In Exercises 5-8, a point is moving along the graph of the given hen each edge is (a) 2 centimeters and (b) 10 centimeters? function such that dxldt is 2 centimeters per second. Find dyldt for the given values of x 19. Surface Area The conditions are the same as Exercise 18 Determine how fast the surfa changing when each edge is (a) 2 centimeters and (b) 10 centimeter 20. Volume The formula for the volume of a cone is V Find the rates of change of the volume if dr/dt is 2 inches per minute and h 3r when (a 6 inches and (b 7. y 21. Volume At a sand and gravel plant sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately WRITING ABOUT CONCEPTS three times the altitude. At what rate is the heigh of the pile changing when the pile is 15 feet high? 9. Consider th e li near function y b. If changes at a constant rate, does y change at a constant rate? If so, does 22. Depth A conical tank (w th vertex down s 10 feet across th change at the same rate as x? Explain. top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth 10. In your ow words, state the guidelines for solving related of the water when th e water is 8 feet rate problems 23. Depth A swimming pool is 12 meters long, 6 meters wide 1 meter deep at the shallow end, and 3 meters deep at the deep 11. Find the rate of change of the distance between the origin end (see figure on next page Water is being pumped nto th f dx/di and a moving point n the graph of y pool at L cubic meter per minute. and there is 1 meter of water centimeters per second the deep end. 12. Find the rate of change of the distance between the origin (a) What percent of the pool is filled? f dx/dt and a moving point on the graph of y b) At what ate is th e water leve sing? centimeters per second 13. Area The radius of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area. when (a 8 centimeters and (b 32 centimeters Comment a A Lueth 18 Research Help 6:30 PM 6/4/2013

Show transcribed image text Ed 9th B Blackboarc x h, ed rds.pd Adobe Read Wind 178 1334 100% Tools Sign (b) If 0 is increasing at the rate of radian per minute, find the when rates of change of the area when 0 /6 and 6 m/3 Explain why the rate of change of the area of the triangle is when not constant even though d6/dt is constant. 16. Volume The radius of a sphere is increasing at a rate of when 3 inches per minute a) Find the rates of c hange of the volume when 9 inches when and 36 inches b) Explain why the rate of change of the volume of the sphere when is not constant even though dr/dt is constant. 17. Volume A spherical balloon is inflated with gas at the rate of when 3, y 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is a) 30 centi and (b) 60 centimeters when 4, y 18. Volume All edges of a cube are expanding at a ate of 6 centimeters per second. How fast is e volume c hanging In Exercises 5-8, a point is moving along the graph of the given hen each edge is (a) 2 centimeters and (b) 10 centimeters? function such that dxldt is 2 centimeters per second. Find dyldt for the given values of x 19. Surface Area The conditions are the same as Exercise 18 Determine how fast the surfa changing when each edge is (a) 2 centimeters and (b) 10 centimeter 20. Volume The formula for the volume of a cone is V Find the rates of change of the volume if dr/dt is 2 inches per minute and h 3r when (a 6 inches and (b 7. y 21. Volume At a sand and gravel plant sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately WRITING ABOUT CONCEPTS three times the altitude. At what rate is the heigh of the pile changing when the pile is 15 feet high? 9. Consider th e li near function y b. If changes at a constant rate, does y change at a constant rate? If so, does 22. Depth A conical tank (w th vertex down s 10 feet across th change at the same rate as x? Explain. top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth 10. In your ow words, state the guidelines for solving related of the water when th e water is 8 feet rate problems 23. Depth A swimming pool is 12 meters long, 6 meters wide 1 meter deep at the shallow end, and 3 meters deep at the deep 11. Find the rate of change of the distance between the origin end (see figure on next page Water is being pumped nto th f dx/di and a moving point n the graph of y pool at L cubic meter per minute. and there is 1 meter of water centimeters per second the deep end. 12. Find the rate of change of the distance between the origin (a) What percent of the pool is filled? f dx/dt and a moving point on the graph of y b) At what ate is th e water leve sing? centimeters per second 13. Area The radius of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area. when (a 8 centimeters and (b 32 centimeters Comment a A Lueth 18 Research Help 6:30 PM 6/4/2013

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