121616949-math.134 - 120 Chapter 6 Applications of the...

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120 Chapter 6 Applications of the Derivative . . . . . . . . . . . -→ x y 3 Figure 6.8 Receding airplane. We are interested in the time at which x = 4; at this time we know that 4 2 + 9 = y 2 , so y = 5. Putting together all the information we have we get 2(4)(500) = 2(5) ˙ y. Thus, ˙ y = 400 mph. EXAMPLE 6.15 You are inflating a spherical balloon at the rate of 7 cm 3 /sec. How fast is its radius increasing when the radius is 4 cm? Here the variables are the radius r and the volume V . We know dV/dt , and we want dr/dt . The two variables are related by means of the equation V = 4 πr 3 / 3. Taking the derivative of both sides gives dV/dt = 4 πr 2 ˙ r . We now substitute the values we know at the instant in question: 7 = 4 π 4 2 ˙ r , so ˙ r = 7 π/ 64 cm/sec. EXAMPLE 6.16 Water is poured into a conical container at the rate of 10 cm 3 /sec. The cone points directly down, and it has a height of 30 cm and a base radius of 10 cm. How fast is the water level rising when the water is 4 cm deep (at its deepest point)? The water forms a conical shape within the big cone; its height and base radius and
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Unformatted text preview: volume are all increasing as water is poured into the container. This means that we actually have three things varying with time: the water level h (the height of the cone of water), the radius r of the circular top surface of water (the base radius of the cone of water), and the volume of water V . The volume of a cone is given by V = πr 2 h/ 3. We know dV/dt , and we want dh/dt . At first something seems to be wrong: we have a third variable r whose rate we don’t know. But the dimensions of the cone of water must have the same proportions as those of the container. That is, because of similar triangles, r/h = 10 / 30 so r = h/ 3. Now we can eliminate r from the problem entirely: V = π ( h/ 3) 2 h/ 3 = πh 3 / 27. We take the derivative of both sides and plug in h = 4 and dV/dt = 10, obtaining 10 = (3 π · 4 2 / 27)( dh/dt ). Thus, dh/dt = 90 / (16 π ) cm/sec....
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