Appearance
question:Given that sin(x) = frac{4}{7} and cos(x) = -frac{sqrt{33}}{sqrt{7}}, find the value of cot(x).
answer:The cotangent function can be expressed as the reciprocal of the tangent function: cot(x) = frac{1}{tan(x)} = frac{cos(x)}{sin(x)} Substituting the given values: cot(x) = frac{-frac{sqrt{33}}{sqrt{7}}}{frac{4}{7}} To simplify, multiply the numerator and denominator by sqrt{7} to rationalize the denominator: cot(x) = frac{-sqrt{33} cdot sqrt{7}}{4 cdot sqrt{7}} This simplifies to: cot(x) = frac{-sqrt{33 cdot 7}}{4} Cancelling out the square root of 7 in the numerator: cot(x) = frac{-sqrt{231}}{4} Therefore, the cotangent of x is cot(x) = -frac{sqrt{231}}{4}.
question:Find a point P on the curve represented by the equation (2y^3 + 6(x^2)y - 12(x^2) + 6y = 1) where the tangent line is parallel to the line (y = -x). Given the implicit derivative of the curve as (frac{4x - 2xy}{x^2 + y^2 + 1}), use this information to solve for the coordinates of point P.
answer:To find the point P, we first set the implicit derivative equal to the slope of the tangent line, which is -1, since the line is parallel to (y = -x). Thus, we have: [frac{4x - 2xy}{x^2 + y^2 + 1} = -1] Given that (y = -x), we substitute (y) with (-x) in the equation: [frac{4x - 2x(-x)}{x^2 + (-x)^2 + 1} = -1] Simplify the equation: [frac{4x + 2x^2}{2x^2 + 1} = -1] Multiplying both sides by the denominator to clear the fraction: [4x + 2x^2 = -2x^2 - 1] Combine like terms: [4x^2 + 4x + 1 = 0] This is a quadratic equation in (x), which can be solved by factoring: [(2x + 1)^2 = 0] Solving for (x): [2x + 1 = 0 implies x = -frac{1}{2}] Now, substitute (x) back into (y = -x) to find (y): [y = -left(-frac{1}{2}right) = frac{1}{2}] Therefore, the point P is (left(-frac{1}{2}, frac{1}{2}right)), and the tangent line at this point is parallel to (y = -x).
question:Identify the precipitate that forms when aqueous solutions of manganese(II) sulfate and sodium carbonate are mixed.
answer:The balanced chemical equation for the reaction is: {eq}{rm{MnS}}{{rm{O}}_4}left( {{rm{aq}}} right) + {rm{N}}{{rm{a}}_2}{rm{C}}{{rm{O}}_3}left( {{rm{aq}}} right) to {rm{MnC}}{{rm{O}}_3}left( {rm{s}} right) + {rm{N}}{{rm{a}}_2}{rm{S}}{{rm{O}}_4}left( {{rm{aq}}} right) {/eq} Manganese(II) sulfate and sodium carbonate are both soluble in water, but manganese(II) carbonate is insoluble. Therefore, the precipitate that forms is manganese(II) carbonate (MnCO₃).
question:On January 1, 2017, Concord Company's retained earnings were reported as 2,139,000. In 2017, the company realized that its 2016 depreciation expense was underreported by 361,000. Additionally, in 2017, Concord had a net income of 815,000 and declared dividends of 275,000. The applicable tax rate is 35%. Calculate the ending balance of retained earnings for Concord Company in 2017, considering the adjustments made to the 2017 retained earnings statement.
answer:To determine the ending balance in the retained earnings for 2017, we'll account for the adjustments as follows: 1. Adjust for the underreported depreciation from 2016. 2. Calculate the additional income tax due because of the underreported depreciation. 3. Subtract the dividends declared. Computation for Ending Retained Earnings: | Particulars | Amount | | --- | --- | | Beginning Retained Earnings | 2,139,000 | | Add: Net Income | 815,000 | | Add: Corrected Depreciation Expense | 361,000 | | Less: Income Tax on Adjusted Depreciation (361,000 x 35%) | 126,350 | | Less: Declared Dividends | 275,000 | | Ending Retained Earnings | 2,444,350 | Hence, the ending balance in the retained earnings for Concord Company as of December 31, 2017, would be 2,444,350.