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- 1. 2.6 Related Rates Don’t get
- 2. Ex. Two rates that are related. Related rate problems are differentiated with respect to time. So, every variable, except t is differentiated implicitly. Given y = x2 + 3, find dy/dt when x = 1, given that dx/dt = 2. y = x2 + 3 dt dx x dt dy 2= Now, when x = 1 and dx/dt = 2, we have 4)2)(1(2 == dt dy
- 3. Procedure For Solving Related Rate Problems 1. Assign symbols to all given quantities and quantities to be determined. Make a sketch and label the quantities if feasible. 2. Write an equation involving the variables whose rates of change either are given or are to be determined. 3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to t. 4. Substitute into the resulting equation all known values for the variables and their rates of change. Solve for the required rate of change.
- 4. Ex. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When this radius is 4 ft., what rate is the total area A of the disturbed water increasing. Given equation: 2 rA π= Givens: 41 == rwhen dt dr Differentiate: dt dr r dt dA π2= ( )( )412π= dt dA π8= ?= dt dA
- 5. An inflating balloon Air is being pumped into a spherical balloon at the rate of 4.5 in3 per second. Find the rate of change of the radius when the radius is 2 inches. Given: sec/5.4 3 in dt dV = r = 2 in. ?: = dt dr Find Equation: 3 3 4 rV π= Diff. & Solve: dt dr r dt dV 2 4π= dt dr2 245.4 π= .09in/sec= dr dt
- 6. The velocity of an airplane tracked by radar An airplane is flying at an elevation of 6 miles on a flight path that will take it directly over a radar tracking station. Let s represent the distance (in miles)between the radar station and the plane. If s is decreasing at a rate of 400 miles per hour when s is 10 miles, what is the speed of the plane.
- 7. Given: Find: Equation: Solve: 10400 =−= s dt ds ?= dt dx x2 + 62 = s2 dt ds s dt dx x 22 = To find dx/dt, we must first find x when s = 10 836100362 =−=−= sx ( ) ( )( )40010282 −= dt dx mph dt dx 500−= Speed is absolute value of velocity = 500 mph
- 8. A fish is reeled in at a rate of 1 foot per second from a bridge 15 ft. above the water. At what rate is the angle between the line and the water changing when there is 25 ft. of line out? 15 ft. x θ
- 9. Given: Find: Equation: Solve: 1−= dt dx x = 25 ft. h = 15 ft. ?= dt dθ x 15 sin =θ 1 15sin − = xθ ( ) dt dx x dt d 2 15cos − −= θ θ dt dx xdt d θ θ cos 15 2 − = ( )1 25 20 25 15 2 − − = dt dθ sec/ 100 3 rad dt d = θ
- 10. Ex. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple in increasing at a constant rate of 1 foot per second. When this radius is 4 ft., what rate is the total area A of the disturbed water increasing. An inflating balloon Air is being pumped into a spherical balloon at the rate of 4.5 in3 per minute. Find the rate of change of the radius when the radius is 2 inches.

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