Problem 79
Volume of a spherical cap A single slice through a sphere of radius \(r\) produces a cap of the sphere. If the thickness of the cap is \(h,\) then its volume is \(V=\frac{1}{3} \pi h^{2}(3 r-h) .\) Graph the volume as a function of \(h\) for a sphere of radius \(1 .\) For what values of \(h\) does this function make sense?
Problem 80
Walking and rowing Kelly has finished a picnic on an island that is \(200 \mathrm{m}\) off shore (see figure). She wants to return to a beach house that is 600 m from the point \(P\) on the shore closest to the island. She plans to row a boat to a point on shore \(x\) meters from \(P\) and then jog along the (straight) shore to the house. a. Let \(d(x)\) be the total length of her trip as a function of \(x\). Find and graph this function. b. Suppose that Kelly can row at \(2 \mathrm{m} / \mathrm{s}\) and jog at \(4 \mathrm{m} / \mathrm{s}\). Let \(T(x)\) be the total time for her trip as a function of \(x .\) Find and graph \(y=T(x)\) c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land to minimize the total time of her trip. What is that minimum time?
Problem 80
Right-triangle relationships Use a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\tan ^{-1}\left(\frac{x}{\sqrt{9-x^{2}}}\right)\right)$$
Problem 80
A capacitor is a device that stores electrical charge. The charge on a capacitor accumulates according to the function \(Q(t)=a\left(1-e^{-t / c}\right),\) where \(t\) is measured in seconds, and \(a\) and \(c>0\) are physical constants. The steady-state charge is the value that \(Q(t)\) approaches as \(t\) becomes large. a. Graph the charge function for \(t \geq 0\) using \(a=1\) and \(c=10\) Find a graphing window that shows the full range of the function. b. Vary the value of \(a\) while holding \(c\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(a ?\) c. Vary the value of \(c\) while holding \(a\) fixed. Describe the effect on the curve. How does the steady-state charge vary with \(c ?\) d. Find a formula that gives the steady-state charge in terms of \(a\) and \(c\)
Problem 81
Optimal boxes Imagine a lidless box with height \(h\) and a square base whose sides have length \(x\). The box must have a volume of \(125 \mathrm{ft}^{3}\). a. Find and graph the function \(S(x)\) that gives the surface area of the box, for all values of \(x>0\) b. Based on your graph in part (a), estimate the value of \(x\) that produces the box with a minimum surface area.
Problem 81
Determine whether the following statements are true and give an explanation or counterexample. a. The range of \(f(x)=2 x-38\) is all real numbers. b. The relation \(y=x^{6}+1\) is not a function because \(y=2\) for both \(x=-1\) and \(x=1\). c. If \(f(x)=x^{-1},\) then \(f(1 / x)=1 / f(x)\). d. In general, \(f(f(x))=(f(x))^{2}\). e. In general, \(f(g(x))=g(f(x))\). f. By definition, \(f(g(x))=(f \circ g)(x)\). g. If \(f(x)\) is an even function, then \(c f(a x)\) is an even function, where \(a\) and \(c\) are nonzero real numbers. h. If \(f(x)\) is an odd function, then \(f(x)+d\) is an odd function, where \(d\) is a nonzero real number. i. If \(f\) is both even and odd, then \(f(x)=0\) for all \(x\).
Problem 81
The height in feet of a baseball hit straight up from the ground with an initial velocity of \(64 \mathrm{ft} / \mathrm{s}\) is given by \(h=f(t)=64 t-16 t^{2},\) where \(t\) is measured in seconds after the hit. a. Is this function one-to-one on the interval \(0 \leq t \leq 4 ?\) b. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels upward. Express your answer in the form \(t=f^{-1}(h)\) c. Find the inverse function that gives the time \(t\) at which the ball is at height \(h\) as the ball travels downward. Express your answer in the form \(t=f^{-1}(h)\) d. At what time is the ball at a height of \(30 \mathrm{ft}\) on the way up? e. At what time is the ball at a height of \(10 \mathrm{ft}\) on the way down?
Problem 82
Using words and figures, explain why the range of \(f(x)=x^{n},\) where \(n\) is a positive odd integer, is all real numbers. Explain why the range of \(g(x)=x^{n},\) where \(n\) is a positive even integer, is all nonnegative real numbers.
Problem 82
Composition of polynomials Let \(f\) be an \(n\) th-degree polynomial and let \(g\) be an \(m\) th-degree polynomial. What is the degree of the following polynomials? a. \(f \cdot f\) b. \(f \circ f\) c. \(f^{*} g\) d. \(f \circ g\).
Problem 83
Use the definition of absolute value to graph the equation \(|x|-|y|=1 .\) Use a graphing utility to check your work.
