answer:Given data: - Mass of the crate, {eq}m = 8.5 kg {/eq} - Force applied, {eq}F = 350 rm N {/eq} - Spring compression, {eq}x = 82.0 cm = 0.82 m {/eq} The spring constant, {eq}k {/eq}, can be found using Hooke's Law: {eq}begin{align*} F &= kx k &= frac{F}{x} k &= frac{350}{0.82} k &= 426.83 N/m end{align*} {/eq} The velocity of the crate when it leaves the spring, {eq}v {/eq}, is determined using the conservation of mechanical energy: {eq}begin{align*} frac{1}{2}mv^2 &= frac{1}{2}kx^2 v &= xsqrt{frac{k}{m}} v &= 0.82 times sqrt{frac{426.83}{8.5}} v &= 5.82 m/s end{align*} {/eq} The distance, {eq}d {/eq}, the crate will slide before stopping can be calculated using the work-energy theorem, considering the frictional force: {eq}begin{align*} frac{1}{2}mv^2 - 0 &= -mu_{k} mg times d quad (text{Final velocity is 0 when it stops}) frac{1}{2}(5.82^2) &= -0.10 times 8.5 times 9.80 times d d &= frac{frac{1}{2}(5.82^2)}{-0.10 times 8.5 times 9.80} d &= frac{33.6366}{-82.95} d &= -17.28 m end{align*} {/eq} However, distance cannot be negative. There must be an error in the calculation. The correct approach for this problem assumes that the initial kinetic energy is converted entirely to potential energy due to friction. The correct formula to find the distance, {eq}d {/eq}, is: {eq}d = frac{v^2}{2mu_{k}g} {/eq} Applying this formula: {eq}begin{align*} d &= frac{5.82^2}{2 times 0.10 times 9.80} d &= frac{33.6366}{19.60} d &= 1.71 m end{align*} {/eq} So, the revised distance the crate will slide before stopping is 1.71 meters.

answer:The function you've defined is equal to the least common multiple of the integers from 1 to x, which can be expressed as: f(x) = text{lcm}(1, 2, 3, ldots, x) This can be understood by considering the prime factorization of each integer from 1 to x. The function f(x) is the product of the highest powers of each prime factor that appears in any of these integers. To see why this is true, consider a prime p and an integer n such that p^n ≤ x. Then p^n will appear as a factor in the prime factorization of every integer from p^n to x. Therefore, the highest power of p that appears in any of these integers is p^n. Now, consider the product of the highest powers of each prime factor that appears in any of the integers from 1 to x. This product is equal to f(x). To see why this is true, let p be a prime factor of x. Then the highest power of p that appears in any of the integers from 1 to x is p^n, where n is the largest integer such that p^n ≤ x. Therefore, p^n appears as a factor in f(x). Since f(x) is the product of the highest powers of each prime factor that appears in any of the integers from 1 to x, it is equal to the least common multiple of these integers.

answer:Systolic pressure measures the pressure in the arteries when the ventricles contract and pump blood into the body, while diastolic pressure measures the pressure in the arteries when the ventricles relax and fill with blood. Systolic pressure is typically higher than diastolic pressure.

answer:The magnitude or norm of a complex number a+bi is given by sqrt{a^2+b^2}. In this case, a=-frac{11}{3} and b=frac{14}{3}, so the magnitude is sqrt{left(-frac{11}{3}right)^2+left(frac{14}{3}right)^2} = frac{sqrt{317}}{3}. The argument or phase angle of a complex number a+bi is given by tan^{-1}left(frac{b}{a}right). However, since the real part of the given complex number is negative, we need to add pi to the result of tan^{-1}left(frac{b}{a}right) to get the argument in the range [-pi, pi]. Therefore, the argument is pi - tan^{-1}left(frac{14}{11}right). Magnitude (Norm): sqrt{left(-frac{11}{3}right)^2+left(frac{14}{3}right)^2} = frac{sqrt{317}}{3} Argument (Phase Angle): tan^{-1}left(frac{frac{14}{3}}{-frac{11}{3}}right) = pi - tan^{-1}left(frac{14}{11}right)