91Ó°ÊÓ

Find the most economical proportions for a tent as in the figure, with no floor.

Given particles of masses m, 2m, and 3m at the points$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{)}$, and $\left(2,3\right)$, find the point P about which their total moment of inertia will be least. (Recall that to find the moment of inertia of m about P, you multiply m by the square of its distance from P.

Find by the Lagrange multiplier method the largest value of the product of three positive numbers if their sum is 1.

Find the two-variable Maclaurin series for the following functions.

cos x sinh y

What proportions will maximize the area shown in the figure (rectangle with isosceles triangles at its ends) if the perimeter is given?

Given $\mathit{f}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\mathbf{0}$and $\mathit{g}\mathbf{(}\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}\mathbf{)}\mathbf{=}\mathbf{0}\mathbf{,}$find a formula for $\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$.

If$\mathit{y}{\mathit{e}}^{\mathbf{x}\mathbf{y}}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$ find$\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$ and${\mathit{d}}^{\mathbf{2}}\mathit{y}\mathbf{/}\mathit{d}{\mathit{x}}^{\mathbf{2}}$ atrole="math" localid="1658830042567" $(0,0)$ .

What proportions will maximize the volume of a projectile in the form of a circular cylinder with one conical end and one flat end, if the surface area is given?

Question: A box has three of its faces in the coordinate planes and one vertex on the plane 2x + 3y + 4z = 6 .Find the maximum value of the box.

If $\mathit{x}{\mathit{s}}^{\mathbf{2}}\mathbf{+}\mathit{y}{\mathit{t}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$and ${\mathit{x}}^{\mathbf{2}}\mathit{s}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathit{t}\mathbf{=}\mathit{x}\mathit{y}\mathbf{-}\mathbf{4}$, find localid="1664251830911" $\frac{\mathbf{∂}\mathbf{x}}{\mathbf{∂}\mathbf{s}}\mathbf{,}\frac{\mathbf{∂}\mathbf{x}}{\mathbf{∂}\mathbf{t}}\mathbf{,}\frac{\mathbf{∂}\mathbf{y}}{\mathbf{∂}\mathbf{s}}\mathbf{,}\frac{\mathbf{∂}\mathbf{y}}{\mathbf{∂}\mathbf{t}}$at $\left(x,y,s,t\right)\mathbf{=}\left(1,-3,2,1\right)$. Hint: To simplify the work, substitute the numerical values just after you have taken differentials.