1 / 15

Related Rates with Area and Volume

Related Rates with Area and Volume. Finding derivative with respect to t. A = s 2. (s, A) . (15, 225) . (10, 100) . (5, 25) . EXAMPLE 1: The side of a square is increasing at a rate of 5cm/s . At what rate is the area changing, when the side is 15 cm long? (A = s 2 ).

shanna
Download Presentation

Related Rates with Area and Volume

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Related RateswithArea and Volume

  2. Finding derivative with respect to t

  3. A = s2 (s, A) (15, 225) (10, 100) (5, 25)

  4. EXAMPLE 1: The sideof a square is increasing at a rate of 5cm/s. At what rate is the area changing, when the side is 15 cm long? (A = s2) s = 5 cm s = 10 cm A = 25 cm2 A = 100 cm2 s = 15 cm A = 225 cm2 A = s2

  5. s = 1 cm V = 1cm3 s = 4 cm V = 64 cm3 s = 7 cm V = 343 cm3 EXAMPLE 2:The edge of a cube is expanding at a rate of 3 cm/s. a) How fast is the volume changing when the edge is 7 cm? V = s3

  6. b) At what rate is the surface area changing when the edge is 7cm? A = 6s2

  7. EXAMPLE 3:An oil tanker ruptures and begins to leak oil in a circular pattern, the radius of which is changing at a rate of 3 m/s. How fast is the area of the spill changing when the radius of the spill has reached 30 m? A = pr2

  8. continued EXAMPLE 4: A sphere is expanding, and the measured rate of increase of its radius is 10 cm/min. a) At what rate is its volume increasing when the radius is 20 cm?

  9. b) At what rate is its surface area increasing when the radius is 10 cm? S.A. = 4p r 2

  10. EXAMPLE 5: A cylindrical tank has a radius of 3 m and a depth of 10 m. It is being filled at a rate of 5 m3/min. How fast is the surface rising? NOTE: As the water rises the height changes but the radius of the water at any level is always 3 m h = 10 m V = p r 2 h Substituter = 3into the formula V =p (3)2h = 9ph Differentiate implicitly

  11. d = 3 m l = 3 m w = 2 m EXAMPLE 6: A rectangular prismatic tank has the following dimensions: length is 3 m, width is 2 m and the depth is 3 m. It is being filled with water, and the surface level is rising at 20cm/minor 0.2 m/min What is the rate of inflow of water to the tank? V = lx wxh NOTE: land w remain constant as water level rises. Substitute l = 3 mandw = 2 m V = (3m)(2m)h = 6(m2)h

  12. EXAMPLE 7: A conical glass vase is being filled with liquid at a rate of 10 cm3/s. The vase is 20 cm high and 3 cm in radius at the top . 20r = 3h Substitute into volume formula: Find the derivative with respect to t

  13. Find the rate at which the water level is rising when the depth is 10 cm.

  14. continued l b a EXAMPLE 8:A water trough on a farm has an isosceles triangular cross section which is 60 cm across the top and 20 cm deep. The trough is 300 cm long. b = 20 cm a = 20 cm l = 300 cm NOTE: so b = 3a V = 0.5 x base of triangle x altitude of triangle x length of trough V = 0.5(3acm)(acm)(300cm) = 450 a 2 cm 3

  15. If it is empty and then is filled at a rate of 500 000 cm3/min, how fast does the water level rise when the deepest point is 9 cm? 500 000 cm3/min = 900(9cm2) = 62 cm/min

More Related