answer:The real solutions to the equation are given by x = frac{1}{4} left(51 - sqrt{2249}right) and x = frac{1}{4} left(51 + sqrt{2249}right).

answer:1. To find the second derivative, we need to differentiate the function twice. 2. The first derivative of the function is: f'(x) = -frac{6}{ sqrt{1-(6x+frac{2}{3})^2}} - frac{14}{3} cos left(frac{14x}{3} + frac{22}{3}right) 3. The second derivative of the function is: f''(x) = frac{196}{9} sin left(frac{2}{3} (7x+11)right) - frac{648 (9x+1)}{left(-324 x^2-72 x+5right)^{3/2}} The answer is f''(x) = frac{196}{9} sin left(frac{2}{3} (7x+11)right) - frac{648 (9x+1)}{left(-324 x^2-72 x+5right)^{3/2}}

answer:Given data: The observed length of the rod is {eq}l' = 0.75;rm m. {/eq} The relative velocity of the observer is {eq}v = 0.5c. {/eq} Standard data: The speed of light in a vacuum is {eq}c = 3 times {10^8};rm {mathop{rm m}nolimits} /s. {/eq} Using the length contraction formula: {eq}l' = lsqrt {1 - frac{{{u^2}}}{{{c^2}}}} {/eq} where {eq}u {/eq} is the velocity of the rod. Solving for {eq}u {/eq}: {eq}begin{align} frac{{l'}}{l} &= sqrt {1 - frac{{{u^2}}}{{{c^2}}}} {left( {frac{{l'}}{l}} right)^2} &= 1 - frac{{{u^2}}}{{{c^2}}} frac{{{u^2}}}{{{c^2}}} &= 1 - {left( {frac{{l'}}{l}} right)^2} u &= csqrt {1 - {{left( {frac{{l'}}{l}} right)}^2}} end{align} {/eq} Since the length of the rod in its rest frame {eq}l {/eq} is unknown, we can use the length contraction formula to relate it to the observed length: {eq}l' = lsqrt {1 - frac{{{v^2}}}{{{c^2}}}} {/eq} Solving for {eq}l {/eq}: {eq}l = frac{{l'}}{{sqrt {1 - frac{{{v^2}}}{{{c^2}}}} }} {/eq} Substituting this into the expression for {eq}u {/eq}: {eq}begin{align} u &= csqrt {1 - {{left( {frac{{l'}}{{frac{{l'}}{{sqrt {1 - frac{{{v^2}}}{{{c^2}}}} }}}} right)}^2}} &= csqrt {1 - {{left( {sqrt {1 - frac{{{v^2}}}{{{c^2}}}} } right)}^2}} &= csqrt {1 - left( {1 - frac{{{v^2}}}{{{c^2}}}} right)} &= csqrt {frac{{{v^2}}}{{{c^2}}}} &= v end{align} {/eq} Therefore, the velocity of the rod as observed by the observer is {eq}v = 0.5c. {/eq}

answer:To determine the amount of purchases needed to cover the membership cost, we can set up the following equation: {eq}displaystyle text{Membership cost} = text{Discounted purchase amount} times text{Discount percentage} {/eq} Substitute the given values: {eq}displaystyle 215 = text{Purchases} times 0.10 {/eq} To find the purchases needed, divide both sides by the discount percentage: {eq}displaystyle text{Purchases } = frac{215}{0.10} {/eq} {eq}displaystyle text{Purchases } = 2,150 {/eq} Therefore, you would need to purchase at least 2,150 worth of brand-name items to cover the cost of the membership.