Problem 56
Industrial costs Dayton Power and Light, Inc. has a power plant on the Miami River where the river is 800 ft wide. To lay a new cable from the plant to a location in the city 2 mi downstream on the opposite side costs \(\$ 180\) per foot across the river and \(\$ 100\) per foot along the land. (a) Suppose that the cable goes from the plant to a point \(Q\) on the opposite side that is \(x\) ft from the point \(P\) directly opposite the plant. Write a function \(C(x)\) that gives the cost of laying the cable in terms of the distance x. (b) Generate a table of values to determine if the least expensive location for point \(Q\) is less than 2000 ft or greater than 2000 \(\mathrm{ft}\) from point \(P .\)
Problem 56
Trigonometric ldentities Let f(x)=\sin x+\cos x (a) Graph \(y=f(x)\) . Describe the graph. (b) Use the graph to identify the amplitude, period, horizontal shift, and vertical shift. (c) Use the formula \(\sin \alpha \cos \beta+\cos \alpha \sin \beta=\sin (\alpha+\beta)\) for the sine of the sum of two angles to confirm your answers.
Problem 57
True or False The function \(f(x)=x^{4}+x^{2}+x\) is an even function. Justify your answer.
Problem 57
Multiple Choice Which of the following is a solution of the equation \(2-3^{-x}=-1 ? \mathrm{}\) (A) \(x=-2 \quad\) (B) \(x=-1 \quad\) (C) \(x=0\) (D) \(x=1 \quad\) (E) There are no solutions.
Problem 57
Tangent Line Consider the circle of radius 5 centered at \((0,0) .\) Find an equation of the line tangent to the circle at the point \((3,4)\) .
Problem 57
Exploration Let y=\sin (a x)+\cos (a x) Use the symbolic manipulator of a computer algebra system (CAS) to help you with the following: (a) Express \(y\) as a sinusoid for \(a=2,3,4,\) and 5 (b) Conjecture another formula for \(y\) for \(a\) equal to any positive integer \(n .\) (c) Check your conjecture with a CAS. (d) Use the formula for the sine of the sum of two angles (see Exercise 56 \(\mathrm{c}\) ) to confirm your conjecture.
Problem 58
True or False The function \(f(x)=x^{-3}\) is an odd function. Justify your answer.
Problem 58
Group Activity Distance From a Point to a Line This activity investigates how to find the distance from a point \(P(a, b)\) to a line \(L : A x+B y=C\) . (a) Write an equation for the line \(M\) through \(P\) perpendicular to \(L\) (b) Find the coordinates of the point \(Q\) in which \(M\) and \(L\) intersect. (c) Find the distance from \(P\) to \(Q\)
Problem 58
Exploration Let \(y=a \sin x+b \cos x\) Use the symbolic manipulator of a computer algebra system (CAS) to help you with the following: (a) Express y as a sinusoid for the following pairs of values: a=2, b=1 ; \quad a=1, b=2 ; \quad a=5, b=2 ; \quad a=2, b=5 a=3, b=4 (b) Conjecture another formula for \(y\) for any pair of positive integers. Try other values if necessary. (c) Check your conjecture with a CAS. (d) Use the following formulas for the sine or cosine of a sum or difference of two angles to confirm your conjecture. \(\begin{aligned} \sin \alpha \cos \beta & \pm \cos \alpha \sin \beta=\sin (\alpha \pm \beta) \\ \cos \alpha \cos \beta \pm \sin \alpha \sin \beta &=\cos (\alpha \mp \beta) \end{aligned}\)
Problem 58
Supporting the Quotient Rule Let \(y_{1}=\ln (x / a), y_{2}=\) \(\ln x, y_{3}=y_{2}-y_{1,}\) and \(y_{4}=e^{y_{3}}\) (a) Graph \(y_{1}\) and \(y_{2}\) for \(a=2,3,4,\) and \(5 .\) How are the graphs of \(y_{1}\) and \(y_{2}\) related? (b) Graph \(y_{3}\) for \(a=2,3,4,\) and \(5 .\) Describe the graphs.
