answer:The area enclosed by a polar curve {eq}r = f(theta){/eq} on the interval {eq}a leq theta leq b{/eq} is given by: {eq}displaystyle A = frac{1}{2} int_{a}^{b} [f(theta)]^2 d theta{/eq} In this case, we have {eq}f(theta) = 3sin theta{/eq} and {eq}a = 0, b = pi/3{/eq}. Substituting these values into the formula, we get: {eq}displaystyle A = frac{1}{2} int_{0}^{pi/3} (3sin theta)^2 d theta{/eq} {eq}displaystyle = frac{9}{2} int_{0}^{pi/3} sin^2 theta d theta{/eq} Using the identity {eq}sin^2 theta = frac{1}{2}(1 - cos 2theta){/eq}, we can rewrite the integral as: {eq}displaystyle A = frac{9}{4} int_{0}^{pi/3} (1 - cos 2theta) d theta{/eq} {eq}displaystyle = frac{9}{4} left[ theta - frac{1}{2} sin 2theta right]_{0}^{pi/3}{/eq} {eq}displaystyle = frac{9}{4} left( frac{pi}{3} - frac{1}{2} cdot frac{sqrt{3}}{2} right){/eq} {eq}displaystyle = frac{3pi}{4} - frac{9sqrt{3}}{16}{/eq} Therefore, the area enclosed by the polar curve is {eq}displaystyle frac{3pi}{4} - frac{9sqrt{3}}{16}{/eq}.

answer:The presence of two non-bonding electron pairs on an oxygen atom causes it to have a trigonal planar geometry, as these electron pairs repel the bonding pairs, leading to a triangular arrangement.

answer:(a) The scatter diagram is as follows: <img>/cimages/multimages/16/x_vs_y8895223669417153337.png</img> (b) The sample mean of x and y are: {eq}bar{x} = 11.3333 {/eq} {eq}bar{y} = 99 {/eq} The sample covariance is: {eq}s_{xy} = 150.8333 {/eq} The equation of the least-squares line is: {eq}hat{y} = 32.1296+5.9003x {/eq}

answer:The divergence of a vector field F = Pmathbf{hat{i}} + Qmathbf{hat{j}} + Rmathbf{hat{k}} is given by the formula text{div} F = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z} So, for the given vector field, we have begin{split} text{div} F &= frac{partial}{partial x} left[ cos^{-1}(x) right] + frac{partial}{partial y} left[ -sin(x-y) right] + frac{partial}{partial z} left[ cos^{-1}(z) right] &= -frac{1}{sqrt{1-x^2}} + cos(x-y) - frac{1}{sqrt{1-z^2}} end{split} The answer is -frac{1}{sqrt{1-x^2}} + cos(x-y) - frac{1}{sqrt{1-z^2}}