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Chapter 2: Problem 97
Find the value of \(k\) such that \(x-4\) is a factor of \(x^{3}-k x^{2}+2 k x-8.\)
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True or False? In Exercises 83 and \(84,\) determine whether the statement is true or false. Justify your answer. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is the entire set of real numbers.
Compound Interest In Exercises 15 and \(16,\) determine the time necessary for \(P\) dollars to double when it is invested at interest rate \(r\) compounded (a) annually, (b) monthly, (c) daily, and (d) continuously. $$r=10 \%$$
Think About It, determine (if possible) the zeros of the function \(g\) when the function \(f\) has zeros at \(x=r_{1}, x=r_{2},\) and \(x=r_{3}\). $$g(x)=f(x-5)$$
pH Levels In Exercises \(51-56\) , use the acidity model given by \(p \mathbf{H}=-\log \left[\mathbf{H}^{+}\right],\) where acidity \((\mathbf{p} \mathbf{H})\) is a measure of the hydrogen ion concentration \(\left[\mathbf{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Find the \(\mathrm{pH}\) when \(\left[\mathrm{H}^{+}\right]=2.3 \times 10^{-5}\)
Page Design A page that is \(x\) inches wide and \(y\) inches high contains 30 square inches of print. The top and bottom margins are each 1 inch deep, and the margins on each side are 2 inches wide (see figure). (a) Write a function for the total area \(A\) of the page in terms of \(x .\) (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper is used.
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