Chapter 4: Problem 12
\(\sqrt[3]{26}\)
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Draining Hemispherical Reservoir Water is flowing at the rate of 6 \(\mathrm{m}^{3} / \mathrm{min}\) from a reservoir shaped like a hemispherical bowl of radius \(13 \mathrm{m},\) shown here in profile. Answer the following questions given that the volume of water in a hemispherical bowl of radius \(R\) is \(V=(\pi / 3) y^{2}(3 R-y)\) when the water is \(y\) units deep. (a) At what rate is the water level changing when the water is 8 m deep? (b) What is the radius \(r\) of the water's surface when the water is \(y\) m deep? (c) At what rate is the radius \(r\) changing when the water is 8 \(\mathrm{m}\) deep?
Moving Shadow A man 6 ft tall walks at the rate of 5 \(\mathrm{ft} / \mathrm{sec}\) toward a streetlight that is 16 \(\mathrm{ft}\) above the ground. At what rate is the length of his shadow changing when he is 10 \(\mathrm{ft}\) from the base of the light?
Production Level Suppose \(c(x)=x^{3}-20 x^{2}+20,000 x\) is the cost of manufacturing \(x\) items. Find a production level that will minimize the average cost of making \(x\) items.
Geometric Mean The geometric mean of two positive numbers \(a\) and \(b\) is \(\sqrt{a b}\) . Show that for \(f(x)=1 / x\) on any interval \([a, b]\) of positive numbers, the value of \(c\) in the conclusion of the Mean Value Theorem is \(c=\sqrt{a b} .\)
Moving Ships Two ships are steaming away from a point \(O\) along routes that make a \(120^{\circ}\) angle. Ship \(A\) moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yards). Ship \(B\) moves at 21 knots. How fast are the ships moving apart when \(O A=5\) and \(O B=3\) nautical miles? 29.5 knots
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