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Related Rates with Area and Volume

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1 Related Rates with Area and Volume
Unit 4 Lesson #5 Related Rates Area and Volume Related Rates with Area and Volume

2 Finding derivative with respect to t
Unit 4 Lesson #5 Related Rates Area and Volume Finding derivative with respect to t 𝒚=𝝅 𝒓 𝟐 𝒙 𝟐 + 𝒚 𝟐 = 𝒛 𝟐 𝒅𝒚 𝒅𝒕 =𝟐𝝅𝒓 𝒅𝒓 𝒅𝒕 𝟐𝒙 𝒅𝒙 𝒅𝒕 +𝟐𝒚 𝒅𝒚 𝒅𝒕 =𝟐𝒛 𝒅𝒛 𝒅𝒕 𝒙 𝒅𝒙 𝒅𝒕 +𝒚 𝒅𝒚 𝒅𝒕 =𝒛 𝒅𝒛 𝒅𝒕

3 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 1: The side of a square is increasing at a rate of 5 cm/s. At what rate is the area changing, when the side is 5 cm long? A = s2 𝒅 𝒅𝒕 𝑨 = 𝒅 𝒅𝒕 𝒔 𝟐 s = 5 cm 𝒅𝑨 𝒅𝒕 =𝟐𝒔 𝒅𝒔 𝒅𝒕 A = 25 cm 2 𝒅𝑨 𝒅𝒕 =𝟐(𝟓𝒄𝒎) 𝟓𝒄𝒎/𝒔 𝒅𝑨 𝒅𝒕 =𝟓𝟎 𝒄𝒎 𝟐 /𝒔

4 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 1: The side of a square is increasing at a rate of 5 cm/s. At what rate is the area changing, when the side is 10 cm long? A = s2 𝒅 𝒅𝒕 𝑨 = 𝒅 𝒅𝒕 𝒔 𝟐 𝒅𝑨 𝒅𝒕 =𝟐𝒔 𝒅𝒔 𝒅𝒕 s = 10 cm 𝒅𝑨 𝒅𝒕 =𝟐(𝟏𝟎𝒄𝒎) 𝟓𝒄𝒎/𝒔 A = 100 cm2 𝒅𝑨 𝒅𝒕 =𝟏𝟎𝟎 𝒄𝒎 𝟐 /𝒔

5 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 1: The side of a square is increasing at a rate of 5 cm/s. At what rate is the area changing, when the side is 15 cm long? A = s2 𝒅 𝒅𝒕 𝑨 = 𝒅 𝒅𝒕 𝒔 𝟐 𝒅𝑨 𝒅𝒕 =𝟐𝒔 𝒅𝒔 𝒅𝒕 𝒅𝑨 𝒅𝒕 =𝟐(𝟏𝟓𝒄𝒎) 𝟓 𝒄𝒎/𝒔 s =15 cm 𝒅𝑨 𝒅𝒕 =𝟏𝟓𝟎 𝒄𝒎 𝟐 /𝒔 A = 225 cm2

6 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 2: The edge of a cube is expanding at a rate of 3 cm/s. How fast is the volume changing when the edge is 1 cm? V = s3 s = 1 cm V = 1 cm3 𝒅 𝒅𝒕 𝑽 = 𝒅 𝒅𝒕 𝒔 𝟑 𝒅𝑽 𝒅𝒕 =𝟑 𝒔 𝟐 𝒅𝒔 𝒅𝒕 𝒅𝑽 𝒅𝒕 =𝟑 𝟏 𝒄𝒎 𝟐 𝟑 𝒄𝒎/𝒔 𝒅𝑽 𝒅𝒕 =𝟗 𝒄𝒎 𝟑 /𝒔

7 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 2: The edge of a cube is expanding at a rate of 3 cm/s. How fast is the volume changing when the edge is 4 cm? V = s3 s = 4 cm V = 64 cm3 𝒅 𝒅𝒕 𝑽 = 𝒅 𝒅𝒕 𝒔 𝟑 𝒅𝑽 𝒅𝒕 =𝟑 𝒔 𝟐 𝒅𝒔 𝒅𝒕 𝒅𝑽 𝒅𝒕 =𝟑 𝟒 𝒄𝒎 𝟐 𝟑 𝒄𝒎/𝒔 𝒅𝑽 𝒅𝒕 =𝟏𝟒𝟒 𝒄𝒎 𝟑 /𝒔

8 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 2: The edge of a cube is expanding at a rate of 3 cm/s. How fast is the volume changing when the edge is 7 cm? V = s3 s = 7 cm V = 343 cm3 𝒅 𝒅𝒕 𝑽 = 𝒅 𝒅𝒕 𝒔 𝟑 𝒅𝑽 𝒅𝒕 =𝟑 𝒔 𝟐 𝒅𝒔 𝒅𝒕 𝒅𝑽 𝒅𝒕 =𝟑 𝟕𝒄𝒎 𝟐 𝟑 𝒄𝒎/𝒔 𝒅𝑽 𝒅𝒕 =𝟒𝟒𝟏 𝒄𝒎 𝟑 /𝒔

9 Unit 4 Lesson #5 Related Rates Area and Volume
b) At what rate is the surface area changing A = 6s2 when the edge is 4 cm? 𝒅 𝒅𝒕 𝑨 = 𝒅 𝒅𝒕 𝟔 𝒔 𝟐 𝒅𝑨 𝒅𝒕 =𝟏𝟐(𝟒 𝒄𝒎) 𝟑 𝒄𝒎/𝒔 𝒅𝑨 𝒅𝒕 =𝟏𝟒𝟒 𝒄𝒎 𝟐 /𝒔 𝒅𝑨 𝒅𝒕 =𝟏𝟐𝒔 𝒅𝒔 𝒅𝒕 when the edge is 1 cm? when the edge is 7 cm? 𝒅𝑨 𝒅𝒕 =𝟏𝟐(𝟏𝒄𝒎) 𝟑 𝒄𝒎/𝒔 𝒅𝑨 𝒅𝒕 =𝟏𝟐(𝟕 𝒄𝒎) 𝟑 𝒄𝒎/𝒔 𝒅𝑨 𝒅𝒕 =𝟐𝟓𝟐 𝒄𝒎 𝟐 /𝒔 𝒅𝑨 𝒅𝒕 =𝟑𝟔 𝒄𝒎 𝟐 /𝒔

10 Unit 4 Lesson #5 Related Rates Area and Volume
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 = p r2 𝒅 𝒅𝒕 𝑨 = 𝒅 𝒅𝒕 𝝅𝒓 𝟐 𝒅𝑨 𝒅𝒕 =𝟐𝝅𝒓 𝒅𝒓 𝒅𝒕 𝒅𝑨 𝒅𝒕 =𝟐𝝅(𝟑𝟎 𝒎) 𝟑 𝒎/𝒔 𝒅𝑨 𝒅𝒕 =𝟏𝟖𝟎𝝅 𝒎 𝟐 /𝒔

11 Unit 4 Lesson #5 Related Rates Area and Volume
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? 𝑽= 𝟒 𝟑 𝝅 𝒓 𝟑 𝒅 𝒅𝒕 𝑽 = 𝒅 𝒅𝒕 𝟒 𝟑 𝝅𝒓 𝟑 𝒅𝑽 𝒅𝒕 =𝟒𝝅 𝒓 𝟐 𝒅𝒓 𝒅𝒕 𝒅𝑽 𝒅𝒕 =𝟒𝝅 𝟐𝟎 𝒄𝒎 𝟐 𝟏𝟎 𝒄𝒎/𝒎𝒊𝒏 𝒅𝑽 𝒅𝒕 =𝟏𝟔𝟎𝟎𝟎𝝅 𝒄𝒎 𝟑 /𝒎𝒊𝒏 continued

12 Unit 4 Lesson #5 Related Rates Area and Volume
b) At what rate is its surface area increasing when the radius is 10 cm? S.A. = 4p r 2 𝒅𝑨 𝒅𝒕 =𝟖𝝅𝒓 𝒅𝒓 𝒅𝒕 𝒅𝑨 𝒅𝒕 =𝟖𝝅(𝟐𝟎 𝒄𝒎) 𝟏𝟎 𝒄𝒎/𝒎𝒊𝒏 𝒅𝑨 𝒅𝒕 =𝟏𝟔𝟎𝟎𝝅 𝒄𝒎 𝟐 /𝒎𝒊𝒏

13 Unit 4 Lesson #5 Related Rates Area and Volume
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 Substitute r = 3 into the formula V = p (3)2 h = 9 p h Differentiate implicitly 𝒅𝑽 𝒅𝒕 =𝟗𝝅 𝒅𝒉 𝒅𝒕 𝟓=𝟗𝝅 𝒅𝒉 𝒅𝒕 𝒅𝒉 𝒅𝒕 = 𝟓 𝒎 𝟑 /𝒎𝒊𝒏 𝟗𝝅 𝒎 𝟐 = 𝟓 𝟗𝝅 𝒎/𝒎𝒊𝒏

14 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 6: A rectangular prismatic tank has the following dimensions: length is 3 m, width is 2 m and the depth is 3 m. w = 2 m d = 3 m l = 3 m It is being filled with water, and the surface level is rising at 20 cm/min or 0.2 m/min What is the rate of inflow of water to the tank? V = l x w x h NOTE: l and w remain constant as water level rises. Substitute l = 3 m and w = 2 m V = (3m)(2m)h = 6(m2)h 𝒅𝑽 𝒅𝒕 =𝟔 𝒅𝒉 𝒅𝒕 =𝟔 𝒎 𝟐 𝟎.𝟐𝒎/𝒎𝒊𝒏 =𝟏.𝟐 𝒎 𝟑 /𝒎𝒊𝒏

15 Unit 4 Lesson #5 Related Rates Area and Volume
EXAMPLE 7: A conical vase is 20 cm high and has a radius of 3 cm at the top . The vase is being filled with liquid at a rate of 10 cm3/s. How fast is height changing? 𝒅𝑽 𝒅𝒕 =𝟏𝟎 cm3/s We know the volume is changing so 𝟑 𝒄𝒎 We are asked how fast the height is changing so we need to determine 𝒅𝒉 𝒅𝒕 𝒓 𝒄𝒎 𝟐𝟎 𝒄𝒎 Since the radius (r), height (h) and slant edge of the vase form similar triangles as the liquid rises we can compare radius to height as a proportion. 𝒉 𝒄𝒎 𝒓 𝒉 = 𝟑 𝟐𝟎 𝒓= 𝟑𝒉 𝟐𝟎 20r = 3h 𝑽= 𝟏 𝟑 𝝅 𝒓 𝟐 𝒉 We can now rewrite the volume formula in terms of V and h only.

16 Unit 4 Lesson #5 Related Rates Area and Volume
𝑽= 𝟏 𝟑 𝝅 𝒓 𝟐 𝒉 Substitute into volume formula: 𝑽= 𝟏 𝟑 𝝅 𝟑𝒉 𝟐𝟎 𝟐 𝒉 𝒓= 𝟑𝒉 𝟐𝟎 𝑽= 𝟏 𝟑 × 𝝅 𝟏 × 𝟗 𝒉 𝟐 𝟒𝟎𝟎 × 𝒉 𝟏 𝑽= 𝟑𝝅 𝒉 𝟑 𝟒𝟎𝟎 Find the derivative with respect to t 𝒅𝑽 𝒅𝒕 = 𝟗𝝅 𝒉 𝟐 𝟒𝟎𝟎 𝒅𝒉 𝒅𝒕

17 𝒅𝑽 𝒅𝒕 = 𝟗𝝅 𝟏𝟎 𝟐 𝟒𝟎𝟎 𝒅𝒉 𝒅𝒕 𝒅𝒉 𝒅𝒕 =𝟏𝟎× 𝟒𝟎𝟎 𝟗𝟎𝟎𝝅 = 𝟒𝟎 𝟗𝝅 𝒄𝒎/𝒔
Find the rate at which the water level is rising when the depth is 10 cm. 𝒅𝑽 𝒅𝒕 = 𝟗𝝅 𝟏𝟎 𝟐 𝟒𝟎𝟎 𝒅𝒉 𝒅𝒕 𝒅𝒉 𝒅𝒕 =𝟏𝟎× 𝟒𝟎𝟎 𝟗𝟎𝟎𝝅 = 𝟒𝟎 𝟗𝝅 𝒄𝒎/𝒔 𝟏𝟎= 𝟗𝝅 𝟏𝟎 𝟐 𝟒𝟎𝟎 𝒅𝒉 𝒅𝒕 𝒅𝒉 𝒅𝒕 =𝟏.𝟒 𝒄𝒎/𝒔 𝒅𝒉 𝒅𝒕 =𝟏𝟎÷ 𝟗𝟎𝟎𝝅 𝟒𝟎𝟎

18 Unit 4 Lesson #5 Related Rates Area and Volume
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. If it is empty and then is filled at a rate of cm3/min, how fast does the water level rise when the deepest point is 9 cm? We know 𝑑𝑉 𝑑𝑡 = cm3/min a l b We need 𝑑𝑎 𝑑𝑡 =? Length stays constant at 300 cm 𝒃 𝒂 = 𝟔𝟎 𝟐𝟎 NOTE: b = 60 cm a = 20 cm l = 300 cm 20b = 60a b = 3a continued

19 Unit 4 Lesson #5 Related Rates Area and Volume
b b = 60 cm a = 20 cm l = 300 cm V = area of triangle end x length of trough V = 0.5 x base of triangle x altitude of triangle x length of trough l = 300 V = 0.5 (b) (a) (l) b = 3a V = 0.5(3a cm)(a cm)(300 cm) = 450 a 2 cm 3 continued

20 Unit 4 Lesson #5 Related Rates Area and Volume
If it is empty and then is filled at a rate of cm3/min, how fast does the water level rise when the deepest point is 9 cm? V = 450 a 2 𝒅𝑽 𝒅𝒕 =𝟗𝟎𝟎𝒂 𝒅𝒂 𝒅𝒕 cm3/min = 900(9cm2) 𝒅𝒂 𝒅𝒕 𝒅𝒂 𝒅𝒕 = 62 cm/min The water level is rising at a rate of 62 cm/min

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