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question:Find the determinant of the matrix before it was converted to reduced row echelon form.
answer:-518400
question:Explain how to determine the salt produced in a neutralization reaction and write the balanced chemical equation for the reaction between Mg(OH)₂ and HBr.
answer:To determine the salt produced in a neutralization reaction, identify the metal ion from the base (Mg²⁺ in this case) and the anion from the acid (Br⁻). Use the criss-cross rule to form the ionic compound, resulting in MgBr₂ as the salt. Balanced chemical equation: {eq}Mg(OH)_2 + 2HBr to MgBr_2 + 2H_2O {/eq}
question:Evaluate the triple integral in cylindrical coordinates: int int int_T z^2 dV where T is the region bounded by the cylinder r = 1 and the planes z = 0 and z = 2.
answer:In cylindrical coordinates, we have dV = r,dr,dtheta,dz. The limits of integration are: 0 leq r leq 1 0 leq theta leq 2pi 0 leq z leq 2 Therefore, the integral becomes: begin{align*} int int int_T z^2 dV &= int_0^{2pi} int_0^1 int_0^2 z^2,r,dz,dr,dtheta &= int_0^{2pi} int_0^1 left[frac{z^3}{3}right]_0^2,r,dr,dtheta &= int_0^{2pi} int_0^1 frac{8}{3}r,dr,dtheta &= int_0^{2pi} left[frac{4}{3}r^2right]_0^1,dtheta &= int_0^{2pi} frac{4}{3},dtheta &= left[frac{4}{3}thetaright]_0^{2pi} &= frac{8pi}{3} end{align*}
question:Captain Daniel, aboard the H.M.S. Khan, is two furlongs away from the dread pirate Nadia and her band of thieves. The Captain has a dfrac{2}{5} chance of hitting the pirate ship with his cannon, while the pirate, who only has one good eye, has a dfrac{1}{5} chance of hitting the Captain's ship. If both ships fire their cannons simultaneously, what is the probability that the Captain hits the pirate ship, but the pirate misses?
answer:Since the events of the Captain hitting the pirate ship and the pirate missing are independent, we can multiply the probabilities of each event occurring to find the probability of both events occurring. The probability that the Captain hits the pirate ship is dfrac{2}{5}. The probability that the pirate misses the Captain's ship is 1 - dfrac{1}{5} = dfrac{4}{5}. Therefore, the probability that the Captain hits the pirate ship, but the pirate misses is dfrac{2}{5} cdot dfrac{4}{5} = dfrac{8}{25}.