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Related rate calculus problems11/21/2023 ![]() ![]() The dimensions of the tank and the rate of fill are all in consistent units. t ^, the rate of change of the height of the water, where t ^ is the moment when the volume is k times the volume of the tank, 0 Helium is pumped into a spherical balloon at the constant rate of 25 cu ft per min.Īt what rate is the surface area of the balloon increasing at the moment when its radius is 8 ft?Ī right-circular conical tank, whose cross-section through its axis is shown in Figure 3.6.5(a), is being filled with water at the constant rate &lambda. How fast is the sand pouring from the hopper? When the height of the pile is observed to be 20 ft, the radius of the base of the pile appears to be increasing at the rate of a foot every two minutes. Sand pouring from a hopper at a steady rate forms a conical pile whose height is observed to remain twice the radius of the base of the cone. Assuming it travels at constant speed, how fast is the north-bound ship traveling? An hour later, a second ship sets sail due north, and at 11:00 PM, the distance between the ships is observed to be increasing at a rate of 97 / 7 knots. ![]() At what rate is the distance between these two ships increasing at 7:00 PM?Īt 1:00 PM a ship traveling at 9 knots sets sail north-east along a line that makes a 30 ° angle with a line running due east. See the Examples 3.6.1-5 for illustrations of this "feature" of most such problems.Īt 1:00 PM a ship sets sail due north at a speed of 14 knots, and an hour later a second ship sets sail due east at a speed of 19 knots. ![]() The time at which the measurement to be made is coincident with the state of one of the variables in the problem, and it is the value of that variable that is used to fix the value of the related rate. Almost never is this magic moment given, or need it be found. One last tip: The typical related-rate problem asks for a "related rate" at some specific moment. Again, the challenge is extracting the formula y x from the words of the statement of the problem. That's all the calculus involved in the typical related-rate problem. Of course, that induces a related change in y, and thus, this change in y is called a "related rate." In this simply stated example, y = y x t, and the rate of change of y, namely, dy dt, is given by the chain rule as dy dx ⋅ dx dt. Varying quantities linked by some quantitative relationship give rise to "related rates of change." The calculus content of the typical such problem is minimal the challenge in a related-rates problem, which is typically a "word problem," is expressing the relationship in terms of the appropriate variables.įor example, suppose the quantity y depends on the quantity x, and that x in turn changes in time, t. ![]() Related-rates problems are exercises designed to illustrate the use of the chain rule. In Equation ( 4.1.2) we see this is when cos 2 θ is largest this is when cos θ = 1, or when θ = 0.Chapter 3: Applications of Differentiation Common sense tells us this is when the car is directly in front of the camera (i.e., when θ = 0). We want to know the fastest the camera has to turn. Now take the derivative of both sides of Equation ( 4.1.1) using implicit differentiation: We need to convert the measurements so they use the same units rewrite -100 mph in terms of ft/s:ĭ x d t = - 100 m hr = - 100 m hr ⋅ 5280 ft m ⋅ 1 3600 hr s = - 146. Letting x represent the distance the car is from the point on the road directly in front of the camera, we haveĪs the car is moving at 100 mph, we have d x d t = - 100 mph (as in the last example, since x is getting smaller as the car travels, d x d t is negative). Figure 4.1.2 suggests we use a trigonometric equation. ![]()
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