. Differential and integral calculus. gone inch a second, at what rate is the volume changing ? Atwhat rate is its convex surface changing ? Ans. 81 it cu. in. a second, |tt V17 sq. in. a second. 12. A reservoir in the form of an inverted conical frustum,radius of smaller base =100 ft. and elements inclined 450 tohorizon, is used to supply an adjacent town with water. If thedepth of the water at any instant is 10 feet, and is decreasing at the rate of two feeta day, at what rate isthe town being sup-plied ? Let AC = x, OC =a\ then 0C=a-\- y = volume ; then y =.— \{a -k x)2 + a2 + (a + x)
. Differential and integral calculus. gone inch a second, at what rate is the volume changing ? Atwhat rate is its convex surface changing ? Ans. 81 it cu. in. a second, |tt V17 sq. in. a second. 12. A reservoir in the form of an inverted conical frustum,radius of smaller base =100 ft. and elements inclined 450 tohorizon, is used to supply an adjacent town with water. If thedepth of the water at any instant is 10 feet, and is decreasing at the rate of two feeta day, at what rate isthe town being sup-plied ? Let AC = x, OC =a\ then 0C=a-\- y = volume ; then y =.— \{a -k x)2 + a2 + (a + x)a)\3 7T = - (3 a2x -f 3 ax2 + *?).3 7T dy = -(3 a2 + 6 ax 4- 3 x2) dx3= 24200 7rcu. ft. a day. 13. Under the action of internal forces a circular cylinder ischanging. When the diameter is 24 in., and increasing at therate of 1 in. a second, the altitude is 48 in., and decreases at therate of 2 in. a second. At what rate is the volume changing ?At what rate is the convex surface changing ? Ans. 288 it cu. in. a second. Not 14. (a) Compare the rates of the ordinate and abscissa ofthe generating point of the logarithmic curve, y = \oga x. (J?) At Analytical Applications 65 what point are the rates equal ? (c) How do the rates comparewhen the moving point crosses the .r-axis ? (a) dy = — dx, , the rate of y is — times the rate of 7 x x (b) At x = i?i. (V) Wheny = o, x = i, .-. dy = mdx. 15. Which increases more rapidly, a number or its logarithm ? Let x = number, then d loga x = — dx ; hence, if #z > ^ the logarithm increases more rapidly ; if m or < than i. 16. (a) How much more rapidly is the number 342 increas-ing than its common logarithm ? ({?) How much more rapidlythan its Napierian logarithm ? (c) If the number increases by1, how much will its common logarithm
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
