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Write the indicated related-rates equation. \(s=\pi r \sqrt{r^{2}+b^{2}} ;\) relate \(\frac{d b}{d t}\) and \(\frac{d r}{d t}\), assuming that \(s\) is constant.

Sketch the graph of a function \(f\) such that all of the following statements are true. \- \(f\) has a relative minimum at \(x=3\). \- \(f\) has a relative maximum at \(x=-1\). \- \(f^{\prime}(x)>0\) for \(x<-1\) and \(x>3\) \- \(f^{\prime}(x)<0\) for \(-1

A ladder 15 feet long leans against a tall stone wall. The bottom of the ladder slides away from the building at a rate of 3 feet per second. a. How quickly is the ladder sliding down the wall when the top of the ladder is 6 feet from the ground? b. At what speed is the top of the ladder moving when it hits the ground?

A girl flying a kite holds the string 4 feet above ground level and lets out string at a rate of 2 feet per second as the kite moves horizontally at an altitude of 84 feet. a. Calculate the rate at which the kite is moving horizontally when 100 feet of string has been let out. b. How far is the kite (above the ground) from the girl at this time?

Boyle's Law for gases states that when the mass of a gas remains constant, the pressure \(p\) and the volume \(v\) of the gas are related by the equation \(p v=c,\) where \(c\) is a constant whose value depends on the gas. Assume that at a certain instant, the volume of a gas is 75 cubic inches and its pressure is 30 pounds per square inch. Because of compression of volume, the pressure of the gas is increasing by 2 pounds per square inch every minute. At what rate is the volume changing at this instant?