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WS22-Related Rates_131_ak_april8

WS22-Related Rates_131_ak_april8 - WS 22 Related Rates...

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WS 22 – Related Rates Name _______________________________ # ____ (4 pages) 1. A pebble is dropped in still water, forming a circular ripple whose radius is expanding at a constant rate of 10 cm/sec. Find a formula giving the area enclosed by the ripple as a function of time. When the radius is 20 cm, how fast is the area enclosed by the ripple increasing? 10 cm/sec = gG ²³´µ = ¶´ · ²³¸µ = ¶¹´³¸µº · | G¼·½ =? gG = 2¶´ = 2¶¹´³¸µº ∗ gG ´ → 20 ² → ¾ G¼·½ = 2¶¹´³¸µº ∗ gG ¾ G¼·½ = 2¶¹20º ∗ 10 = 400¶ ¿À · /Á¿ The area is increasing at a rate of 1,256.64 square centimeters per second when the radius is 20 centimeters. 2. (A) When the radius of a balloon (assume it is a sphere) is 10 cm, at what rate is the volume of the balloon changing with respect to its radius? à → gG =? ó´µ = Å Æ ¶´ Æ ´ → 10 gG = 4¶´ · gG ¾ G¼Ç½ = 4¶³10µ · = 400¶ ¿À · (B) If the radius of the spherical balloon is increasing by 0.5 cm/sec, how fast is the air being blown into the balloon when the radius is 6 cm? à → =? ó¸µ = Å Æ ¶¹´³¸µº Æ ´ → 6 gG = 0.5 = 4¶´ · gG ¾ G¼È = 4¶³6µ · ∗ É Ç · Ê = 72¶ ¿À Æ /Á¿ The volume is increasing at a rate of about 226.2 cubic centimeters per second when the radius is 6 centimeters . 3. The mass of a circular oil slick of radius r is ( ln(1 )) M K r r = - + , where K is a positive constant. What is the relationship between the rate of change of the radius with respect to time and the rate of change of the mass with respect to time? ( ln(1 )) M K r r = - + = Ì ÉÍ − Í ÍÎÏ Ê ∗ gG
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4. (Text 4.6 #16) The volume of a square base pyramid is g = G ± ² ³ .
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