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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 ).

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Related Rates with Area and Volume

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  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

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