Problem Statement: A bowl is a solid of revolution bounded by the surfaces obtained by rotating the curves \[C_1: x^2=4y \quad \text{and} \quad C_2: x^2=8(y-k)\] about the y-axis, where \(k\) is a constant greater than zero. (a) What is the diameter of the mouth of the bowl in terms of \(k\)? (b) Find the capacity... » read more
Problem Statement: We are given the functions \[f(x)=e^x−ax\] and \[g(x)=ax−\ln x\] with \(x>0\) for \(g(x)\) It is given that \(f(x)\) and \(g(x)\) have the same minimum value. (1) Find \(a\) (2) Prove that there exists a line \(y=b\) intersecting both curves \(y=f(x)\) and \(y=g(x)\) at three distinct points whose \(x\)-coordinates form an arithmetic progression Answer: (1) \(f'(x)=e^x-a\) \(f′(x)=0\) ⟹ \(0=e^x-a\) ∴ \(x= \ln a\) The minimum value... » read more
Problem Statement:A box contains 5 balls labeled from 1 to 5. If three balls are drawn with replacement, let X be the number of distinct balls that are drawn at least once in the three draws. Find E(X). Answer: \(P(X=3)=\frac{5}{5}\times\frac{4}{5}\times\frac{3}{5}=\frac{12}{25}\) \(P(X=2)=\frac{5}{5}\times\frac{1}{5}\times\frac{4}{5}\times\text{C}^3_2=\frac{12}{25}\) \(P(X=1)=\frac{5}{5}\times\frac{1}{5}\times\frac{1}{5}=\frac{1}{25}\) \(E(X)=3 \times P(X=3)+2 \times P(X=2)+1 \times P(X=1)\) \(=3\times\frac{12}{25}+2\times\frac{12}{25}+1\times\frac{1}{25}\) \(=\frac{61}{25}\) There are approximately 2.44... » read more