answer:Because {a_n} is a geometric progression, we can use its properties to set up an equation relating a_2, a_6, and a_{10}: a_{6}^2 = a_{2} cdot a_{10}. Plugging in the given values, we have: 162^2 = 2 cdot a_{10}. Now we can solve for a_{10}: a_{10} = frac{162^2}{2} = frac{26244}{2} = 13122. Hence, the value of a_{10} is boxed{13122}.

answer:1. **Amplitude**: The amplitude is the coefficient of the sine function, which in this case is the absolute value of 3. Therefore, the amplitude is boxed{3}. 2. **Period**: The period of a sine function y = a sin(bx + c) is given by frac{2pi}{b}. Here, b = 3, so the period is frac{2pi}{3}. Hence, the period is boxed{frac{2pi}{3}}. 3. **Phase Shift**: The phase shift is calculated from the expression inside the sine function. For y = a sin(bx + c), the phase shift is -frac{c}{b}. In y = 3 sin left( 3x - frac{pi}{4} right), c = -frac{pi}{4} and b = 3. Thus, the phase shift is -left(-frac{pi}{4} / 3right) = frac{pi}{12}. Therefore, the phase shift is boxed{frac{pi}{12}}.

answer:1. **Identify the domain**: The new equation is frac{2x-4}{3x+1} = frac{2x-k}{5x+4}. We exclude x values that make the denominators zero, so x neq -frac{1}{3}, -frac{4}{5}. 2. **Cross-multiply**: [ (2x-4)(5x+4) = (2x-k)(3x+1) ] Expanding both sides, we get: [ 10x^2 - 12x - 16 = 6x^2 - kx + 2x - k ] [ 10x^2 - 12x - 16 = 6x^2 + (2-k)x - k ] 3. **Simplify and rearrange**: [ 4x^2 - (10-k)x - 16 + k = 0 ] 4. **Analyze coefficient of x**: If 10 - k = 0, then k = 10, making the equation: [ 4x^2 - 16 = 0 ] [ x^2 = 4 ] giving x = pm 2, which are valid in the domain. Rechecking for no solution when k = 10, x = pm 2 is satisfactory and the left-hand side becomes zero. 5. **Conclusion**: The equation frac{2x-4}{3x+1} = frac{2x-k}{5x+4} has no solution when k = 10, therefore the correct answer is textbf{(E) 10}. The correct answer is boxed{textbf{(E)} 10}.

answer:First, let's calculate the total amount Mitch needs to reserve for the license, registration, and docking fees. License and registration: 500 Docking fees: 3 times the license and registration = 3 * 500 = 1500 Total reserved amount: 500 (license and registration) + 1500 (docking fees) = 2000 Now, let's subtract the reserved amount from the total savings to find out how much Mitch has left to spend on the boat itself. Total savings: 20000 Reserved amount: 2000 Amount left for the boat: 20000 - 2000 = 18000 The cost of the boat is 1500 per foot. To find out the maximum length of the boat Mitch can afford, we divide the amount he has left by the cost per foot. Maximum length of the boat: 18000 / 1500 per foot = 12 feet So, the longest boat Mitch can buy is boxed{12} feet.