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Chapter 1: Problem 19
Graph each function with a graphing utility using the given window. Then state the domain and range of the function. $$f(x)=\left(9-x^{2}\right)^{3 / 2} ; \quad[-4,4] \times[0,30]$$
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Find a formula for a function describing the given situation. Graph the function and give a domain that makes sense for the problem. Recall that with constant speed. distance \(=\) speed \(\cdot\) time elapsed or \(d=v t\) A function \(y=f(x)\) such that if your car gets \(32 \mathrm{mi} / \mathrm{gal}\) and gasoline costs \(\$ x /\) gallon, then \(\$ 100\) is the cost of taking a \(y\) -mile trip
The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles; one by wrapping a rope around the equator and then a second circle that is \(38 \mathrm{ft}\) longer than the first rope (see figure). How much space is between the ropes?
Make a sketch of the given pairs of functions. Be sure to draw the graphs accurately relative to each other. $$y=x^{3} \text { and } y=x^{7}$$
Use the following steps to prove that \(\log _{b}(x y)=\log _{b} x+\log _{b} y\). a. Let \(x=b^{p}\) and \(y=b^{q}\). Solve these expressions for \(p\) and \(q\) respectively. b. Use property El for exponents to express \(x y\) in terms of \(b, p\) and \(q\). c. Compute \(\log _{b}(x y)\) and simplify.
(Torricelli's law) A cylindrical tank with a cross-sectional area of \(100 \mathrm{cm}^{2}\) is filled to a depth of \(100 \mathrm{cm}\) with water. At \(t=0,\) a drain in the bottom of the tank with an area of \(10 \mathrm{cm}^{2}\) is opened, allowing water to flow out of the tank. The depth of water in the tank at time \(t \geq 0\) is \(d(t)=(10-2.2 t)^{2}\) a. Check that \(d(0)=100,\) as specified. b. At what time is the tank empty? c. What is an appropriate domain for \(d ?\)
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