Higher Concepts of MathematicsCALCULUSExample 1: A stone is dropped into a quiet lake, and waves move in circles outward from thelocation of the splash at a constant velocity of 0.5 feet per second. Determine therate at which the area of the circle is increasing when the radius is 4 feet.Solution:Using the formula for the area of a circle,A pr^{2}take the derivative of both sides of this equation with respect to time t.dAdt2prdrdtBut, dr/dt is the velocity of the circle moving outward which equals 0.5 ft/s anddA /dtis the rate at which the area is increasing, which is the quantity to bedetermined. Set r equal to 4 feet, substitute the known values into the equation,and solve for dA /dt. dAdt2prdrdtdAdt(2)(3.1416)(4 ft)0.5 ft/sdAdt12.6 ft^{2}/sThus, at a radius of 4 feet, the area is increasing at a rate of 12.6 square feet persecond.Example 2:A ladder 26 feet long is leaning against a wall. The ladder starts to move suchthat the bottom end moves away from the wall at a constant velocity of 2 feet persecond. What is the downward velocity of the top end of the ladder when thebottom end is 10 feet from the wall?Solution:Start with the Pythagorean Theorem for a right triangle:a^{2}= c^{2}- b^{2}Rev. 0 Page 39MA-05

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