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30..answer no 7 do like example 6 solution

30..answer no 7 do like example 6 solution

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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30..answer no 7 do like example 6 solution 

TUTORIAL 30 The radius of a semicircle is increasing at the rate of 0.8 cm/s. Calculate the rate of change in the area and the perimeter of the semicircle when the radius is 5 cm 2 A stone dropped into a still pond sends out a circular ripple whose area increases at a rate of 5 cm'/s. How fast is the radius of the ripple changing when the radius is 2 cm ? 3 Oil from a leaking oil tanker radiates outward in the form of a circular film on the surface of the water. If the area of the circle increases at the rate of 400 m'/min, how fast is the radius of the circle increasing when the area is 1225n m. 4. The radius, rcm of a circle at time t is given by r = 21 + 3. Determine the rate of change in the area when t = 1. (Give your answer in terms of n). 5. The area of a semicircle is increasing at the rate of 4n cm s'. Calculate the rate of change in the radius of the semicircle when the radius is 5 cm. 6. Oil spilled from a ruptured tanker spreads in a circle whose radius inreases at a constant rate of 2 cm/s. How fast is the area of the spill increasing after 5 minutes ? 1. A closed cuboid has a square base and its height is twice its base. If x is the length of the base and x increases at a rate of 3 cm s, find the rate of change of the surface area when x is 4 cm. 8. At what rate is the volume and surface area of a cube is changing when the side is 15 cm and each side is increasing at the rate of 0.6 cm/sec. 9. The radius of a spherical balloon increases at a rate of 0.2 cms. How fast is the volume changing at a certain instant when its volume is 288t cm. 10. A spherical balloon is to be deflated at a rate of p cm's'. The radius of the balloon decreases ata rate of 0.5 cms when its volume is 288t cm. Find the value of p. 11. A boy is blowing a spherical balloon at a rate of 50 cm/s. At what rate is the radius of the balloon changing when the radius is 10 cm. 12. The surface area of a sphere decreases at a rate of 5 m's. How fast is the radius changing when the radius is 3 m. 13. If the radius of a sphere is increasing at the constant rate of 2 mm per sec, how fast is the volume changing when the surface area is 8 mm ? 4. (Hint: Vaphiere = r: Agphere = 4nr") Answers: dA 4( 2) de 40
Transcribed Image Text:TUTORIAL 30 The radius of a semicircle is increasing at the rate of 0.8 cm/s. Calculate the rate of change in the area and the perimeter of the semicircle when the radius is 5 cm 2 A stone dropped into a still pond sends out a circular ripple whose area increases at a rate of 5 cm'/s. How fast is the radius of the ripple changing when the radius is 2 cm ? 3 Oil from a leaking oil tanker radiates outward in the form of a circular film on the surface of the water. If the area of the circle increases at the rate of 400 m'/min, how fast is the radius of the circle increasing when the area is 1225n m. 4. The radius, rcm of a circle at time t is given by r = 21 + 3. Determine the rate of change in the area when t = 1. (Give your answer in terms of n). 5. The area of a semicircle is increasing at the rate of 4n cm s'. Calculate the rate of change in the radius of the semicircle when the radius is 5 cm. 6. Oil spilled from a ruptured tanker spreads in a circle whose radius inreases at a constant rate of 2 cm/s. How fast is the area of the spill increasing after 5 minutes ? 1. A closed cuboid has a square base and its height is twice its base. If x is the length of the base and x increases at a rate of 3 cm s, find the rate of change of the surface area when x is 4 cm. 8. At what rate is the volume and surface area of a cube is changing when the side is 15 cm and each side is increasing at the rate of 0.6 cm/sec. 9. The radius of a spherical balloon increases at a rate of 0.2 cms. How fast is the volume changing at a certain instant when its volume is 288t cm. 10. A spherical balloon is to be deflated at a rate of p cm's'. The radius of the balloon decreases ata rate of 0.5 cms when its volume is 288t cm. Find the value of p. 11. A boy is blowing a spherical balloon at a rate of 50 cm/s. At what rate is the radius of the balloon changing when the radius is 10 cm. 12. The surface area of a sphere decreases at a rate of 5 m's. How fast is the radius changing when the radius is 3 m. 13. If the radius of a sphere is increasing at the constant rate of 2 mm per sec, how fast is the volume changing when the surface area is 8 mm ? 4. (Hint: Vaphiere = r: Agphere = 4nr") Answers: dA 4( 2) de 40
Example 6: A cuboid metal slab has a width of x m, length 2x m and height 2 m. 2 m 2x m When heated, the widih expanded at the rate of 0.01 mh. Find the rate of change of the surface area when the volume is 36 m, Solution: dA dx = 0.01. dt = ?, V = 36 dt V = 4x 4x = 36 x = 2 ( 2x + 4x + 2x) 4x + 12x A %3D dA dx dA %3D dt dt dx 0.01 x (8x + 12) %3! 0.01 x (8(3) + 12) 0.36 m2 h %3D Alternative Solution: A = 4x + 12x dA dt dx dx + 12 dt 8x %3D dt 8(3)(0.01) + 12 (0.01) 0.36 m h
Transcribed Image Text:Example 6: A cuboid metal slab has a width of x m, length 2x m and height 2 m. 2 m 2x m When heated, the widih expanded at the rate of 0.01 mh. Find the rate of change of the surface area when the volume is 36 m, Solution: dA dx = 0.01. dt = ?, V = 36 dt V = 4x 4x = 36 x = 2 ( 2x + 4x + 2x) 4x + 12x A %3D dA dx dA %3D dt dt dx 0.01 x (8x + 12) %3! 0.01 x (8(3) + 12) 0.36 m2 h %3D Alternative Solution: A = 4x + 12x dA dt dx dx + 12 dt 8x %3D dt 8(3)(0.01) + 12 (0.01) 0.36 m h
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