answer:First, we can factor out a -9 from the polynomial: -9x^3 - 369x^2 - frac{19125x}{4} - 18900 = -9(x^3 + 41x^2 + frac{2125x}{4} + 2100) Next, we can use the sum of cubes factoring formula to factor the remaining polynomial: x^3 + 41x^2 + frac{2125x}{4} + 2100 = (x + 15)^3 - 125 Therefore, we have: -9x^3 - 369x^2 - frac{19125x}{4} - 18900 = -9(x + 15)^3 + 1125 Finally, we can factor the remaining polynomial using the difference of squares factoring formula: -9(x + 15)^3 + 1125 = -9(x + 15)^2(x + 15 - 16)(x + 15 + 16) Simplifying, we get: -9(x + 15)^2(x - 1)(x + 31) Therefore, the fully factored form of the polynomial is: -9(x + frac{15}{2})(x + 16)(x + frac{35}{2}) The answer is -9(x + frac{15}{2})(x + 16)(x + frac{35}{2})

answer:The first-order Taylor expansion of f(x) around x = 1 can be written as: f(x) approx f(1) + f'(1)(x - 1) For f(x) = frac{1}{16x^4} + arcsin(x), we have: f(1) = frac{1}{16} + frac{pi}{2} And the derivatives evaluated at x = 1 are: f'(x) = -frac{1}{4x^5} + frac{1}{sqrt{1-x^2}} So, f'(1) = -frac{1}{4} + sqrt{2} Substituting these into the Taylor expansion formula, we get: f(x) approx left(frac{1}{16} + frac{pi}{2}right) + left(-frac{1}{4} + sqrt{2}right)(x - 1) Simplifying: f(x) approx frac{1}{16} - frac{1}{4}(x - 1) + sqrt{2}(x - 1) + frac{pi}{2} f(x) approx frac{1}{16} + frac{x - 1}{4} + sqrt{2}(x - 1) + frac{pi}{2} Rearranging terms to combine like terms: f(x) approx frac{x + 1}{4} + sqrt{2}sqrt{x + 1} - frac{pi}{2} + frac{1}{16} This is the first-order Taylor expansion of f(x) around x = 1.

answer:Given: - Maximum extension of the spring (x) = 10.5 cm = 0.105 m - Force on the spring (F) = Weight of the package (mg) - Reading on the balance (m) = 32.8 kg - Acceleration due to gravity (g) = 9.81 m/s² - Oscillation frequency (f) = 2.25 Hz a) Using Hooke's Law, we can find the spring constant (k): [ F = kx ] [ 32.8 kg cdot 9.81 m/s^2 = k cdot 0.105 m ] [ k = frac{32.8 kg cdot 9.81 m/s^2}{0.105 m} ] [ k = 3064.46 N/m ] b) For the spring-mass system, the oscillation frequency is related to the spring constant and the mass (m) as: [ f = frac{1}{2pi} sqrt{frac{k}{m}} ] [ 2.25 = frac{1}{2pi} sqrt{frac{3064.46}{m}} ] Solving for m: [ m = frac{3064.46}{(2pi)^2 cdot f^2} ] [ m = frac{3064.46}{(2pi)^2 cdot (2.25)^2} ] [ m = 15.33 kg ] The weight of the package (W) is given by: [ W = mg ] [ W = 15.33 kg cdot 9.81 m/s^2 ] [ W = 150.42 N ] Thus, the spring constant is 3064.46 N/m, and the package's weight is 150.42 N.

answer:Although maximizing economic surplus can contribute to overall welfare, it may not guarantee equitable distribution of benefits. Policymakers often consider additional factors to ensure societal well-being. An economy with maximum economic surplus might still have pockets of poverty, as some individuals might be left with minimal resources. Therefore, aspects like unemployment rate and welfare programs are crucial to address inequality. Society typically aims to promote economic growth while ensuring a more inclusive distribution of its benefits.