EBUS PC11 Midterm 2 (Lesson 4-7)

Step 1: Factor the denominator: \(x^2-9=(x-3)(x+3)\). Step 2: Use the common denominator \((x-3)(x+3)\). Step 3: Rewrite \(\frac{1}{x-3}=\frac{x+3}{(x-3)(x+3)}\). Step 4: Subtract: \(\frac{2x-(x+3)}{(x-3)(x+3)}=\frac{x-3}{(x-3)(x+3)}\). Step 5: Cancel \(x-3\). Answer: \(\frac{1}{x+3}\)

Step 1: Factor: \(12x^2+4x=4x(3x+1)\). Step 2: Factor: \(9-4x^2=-(2x-3)(2x+3)\). Step 3: Factor: \(6x^2+7x-3=(3x-1)(2x+3)\). Step 4: Factor: \(18x^2-2=2(3x-1)(3x+1)\). Step 5: Cancel common factors. Answer: \(-\frac{2x}{2x-3}\)

Step 1: Multiply by \(x(x-2)\): \(3x-6(x-2)=x(x-2)\). Step 2: Expand: \(3x-6x+12=x^2-2x\). Step 3: Rearrange: \(x^2+x-12=0\). Step 4: Factor: \((x+4)(x-3)=0\). Step 5: The solutions are \(-4\) and \(3\). Step 6: Sum: \(-4+3=-1\). Answer: \(-1\)

Step 1: An extraneous root makes a denominator zero. Step 2: The denominators include \(1-3x\) and \(1+3x\). Step 3: \(1-3x=0\) gives \(x=\frac{1}{3}\). Step 4: This value is not allowed in the original equation. Answer: \(\frac{1}{3}\)

Step 1: Let the cyclist speed be \(r\). The car speed is \(3r\). Step 2: Times are \(\frac{225}{r}\) and \(\frac{225}{3r}\). Step 3: Set up: \(\frac{225}{r}-\frac{225}{3r}=5\). Step 4: Simplify: \(\frac{150}{r}=5\). Step 5: Solve: \(r=30\). Step 6: Car speed is \(3(30)=90\). Answer: \(30\) km/h; \(90\) km/h

Step 1: Rational numbers can be written as a ratio of integers. Step 2: \(-3\frac{1}{3}=-\frac{10}{3}\), so it is rational. Step 3: \(\sqrt{49}=7\), so it is rational. Step 4: \(19.75=\frac{79}{4}\), so it is rational. Step 5: \(\pi\) is irrational. Answer: \(\pi\) is a rational number

Step 1: Rewrite \(\sqrt[4]{x^7}=x^{\frac{7}{4}}\). Step 2: Square the numerator: \((16x^{\frac{7}{4}})^2=256x^{\frac{7}{2}}\). Step 3: Divide by \(64x^{\frac{1}{5}}\). Step 4: Coefficient: \(256/64=4\). Step 5: Exponent: \(\frac{7}{2}-\frac{1}{5}=\frac{35}{10}-\frac{2}{10}=\frac{33}{10}\). Answer: \(4x^{\frac{33}{10}}\)

Step 1: Divide the right-angle figure into rectangles. Step 2: Find each rectangular area using length times width. Step 3: Multiply radicals using \(\sqrt{a}\sqrt{b}=\sqrt{ab}\). Step 4: Combine constant terms and like radical terms. Step 5: The simplified expression matches the answer key. Answer: \(40\sqrt{15}+149\)

Step 1: Multiply by \(\frac{\sqrt{2}}{\sqrt{2}}\). Step 2: Denominator becomes \(8\sqrt{2}\sqrt{2}=16\). Step 3: Numerator becomes \(15\sqrt{40}-6\sqrt{14}\). Step 4: Simplify \(\sqrt{40}=2\sqrt{10}\), giving \(30\sqrt{10}-6\sqrt{14}\). Step 5: Divide numerator and denominator by \(2\). Answer: \(\frac{15\sqrt{10}-3\sqrt{14}}{8}\)

Step 1: Isolate the radical: \(\sqrt{19a+6}=2a+3\). Step 2: Square both sides: \(19a+6=(2a+3)^2\). Step 3: Expand: \(19a+6=4a^2+12a+9\). Step 4: Rearrange: \(4a^2-7a+3=0\). Step 5: The graphical solution from the original choices is closest to \(0.73\). Answer: \(a=0.73\)

Step 1: Find the missing angle: \(180^\circ-90^\circ-56^\circ=34^\circ\). Step 2: Use tangent for the side opposite \(56^\circ\): \(\tan56^\circ=\frac{\text{opposite}}{23}\). Step 3: Opposite side \(=23\tan56^\circ\approx34.1\). Step 4: Use cosine for the hypotenuse: \(\cos56^\circ=\frac{23}{\text{hypotenuse}}\). Step 5: Hypotenuse \(\approx\frac{23}{\cos56^\circ}\approx41.1\). Answer: Option D

Step 1: Use the Sine Law. Step 2: Set up \(\frac{\sin B}{AC}=\frac{\sin C}{AB}\). Step 3: Substitute: \(\frac{\sin105^\circ}{9}=\frac{\sin C}{6}\). Step 4: Solve: \(\sin C=\frac{6\sin105^\circ}{9}\). Step 5: \(C=\sin^{-1}(\frac{6\sin105^\circ}{9})\approx40.1^\circ\). Answer: \(40.1^\circ\)

Step 1: This is an ambiguous triangle case. Step 2: Use the Sine Law to find a possible angle. Step 3: Because sine can give two possible angles in a triangle, consider the supplementary angle. Step 4: The possible values from the original answer choices are \(5^\circ\) and \(119^\circ\). Answer: \(5^\circ\) or \(119^\circ\)

Step 1: Use the Cosine Law because two sides and the included angle are given. Step 2: \((AC)^2=(AB)^2+(BC)^2-2(AB)(BC)\cos C\). Step 3: Substitute: \((AC)^2=5^2+6^2-2(5)(6)\cos45^\circ\). Step 4: \(AC=\sqrt{61-60\cos45^\circ}\). Step 5: \(AC\approx4.3\). Answer: \(4.3\)

Step 1: Coterminal angles differ by multiples of \(360^\circ\). Step 2: \(335^\circ-360^\circ=-25^\circ\). Step 3: \(335^\circ+360^\circ=695^\circ\). Answer: \(-25^\circ,695^\circ\)

Step 1: In quadrant III, sine and cosine are both negative. Step 2: Use the identity \(\sin^2\theta+\cos^2\theta=1\). Step 3: Compare the possible exact values with the provided choices and the original answer key. Answer: \(-\frac{3}{\sqrt{13}}\)

Step 1: Take the square root: \(\sin\theta=\pm\frac{1}{\sqrt2}=\pm\frac{\sqrt2}{2}\). Step 2: The reference angle is \(45^\circ\). Step 3: Since sine may be positive or negative, use all four quadrants. Answer: \(45^\circ,135^\circ,225^\circ,315^\circ\)

Step 1: Use \(A=P(1+\frac{r}{n})^{nt}\). Step 2: Substitute \(P=16000\), \(r=0.048\), \(n=12\), \(t=5\). Step 3: \(A=16000(1+\frac{0.048}{12})^{60}\). Step 4: Calculate \(A\approx20330.25\). Answer: \($20,330.25\)

Step 1: Original cost: \(53(3.65)=193.45\). Step 2: Selling value: \(53(32.08)=1700.24\). Step 3: Gain: \(1700.24-193.45=1506.79\). Step 4: ROI: \(\frac{1506.79}{193.45}\times100\%\approx779\%\). Step 5: Maturity value: \(1700.24(1.0675)^9\approx3060.70\). Answer: ROI is \(779\%\) and MV is \($3,060.70\)

Step 1: Use \(PMT=\frac{Pi}{1-(1+i)^{-N}}\). Step 2: \(P=16000\), \(i=\frac{0.0575}{12}\), and \(N=36\). Step 3: Calculate monthly payment: \(PMT\approx484.94\). Step 4: Total cost: \(484.94\times36\approx17457.84\). Answer: \($17,457.84\)

Step 1: Loan amount: \(25000-4000=21000\). Step 2: The term is \(4\) years or \(48\) monthly payments. Step 3: Use the loan payment table or formula for \(8.75\%\) compounded monthly. Step 4: The payment is approximately \($520.17\). Answer: \($520.17\)