Problems on Finding Greatest and Least Values

Among all rectangles having a given area $S$, find the rectangle:

Find the greatest area of a rectangle inscribed in a circle of radius $R$.

Find the point on the hyperbola $x^{2} / 2-y^{2} = 1$ nearest to the point $(3 ; 0)$.

Find the point on the parabola $y = x^{2}$ nearest to the point $A(2 ; 1 / 2)$.

Find the greatest area of a rectangle two of whose vertices lie on the axes $Ox$ and $Oy$ of a rectangular coordinate system, the third at the point $(0 ; 0)$, and the fourth on the parabola $y = 3-x^{2}$.

Find the slope of the line passing through the point $A(1 ; 2)$ that cuts off from the first coordinate angle a triangle of least area.

Find the length of the lateral side of the trapezoid having the least perimeter among all isosceles trapezoids with given area $S$ and angle $\alpha$ between the lateral side and the lower base.

Lines are drawn through the point $A(2 ; 1 / 4)$ intersecting the positive semi-axes at points $B$ and $C$. Find the equation of the line for which the segment $BC$ has the least length.

Find the acute angles of the right triangle having the greatest area among all triangles for which the sum of the lengths of one leg and the hypotenuse is constant.

Find the least length of a segment that divides an equilateral triangle with side $a$ into two figures of equal area.

Determine the angles of the triangle $ABC$ of greatest area, if the length of its base $BC$ is given and it is known that the angle $BAC$ equals $\alpha$.

Find the sides of the rectangle of greatest area inscribed in the ellipse

such that the sides of the rectangle are parallel to the axes of the ellipse.

Compute the greatest area of a trapezoid inscribed in a semicircle of radius $R$ such that the lower base of the trapezoid is the diameter of the semicircle.

The cross-section of a tunnel has the shape of a rectangle topped by a semicircle. Determine the radius of the semicircle for which the area of the cross-section is greatest, if the perimeter of the cross-section equals $p$.

Through which point of the ellipse $x^{2} / a^{2}+y^{2} / b^{2} = 1$ should a tangent be drawn so that the area of the triangle formed by this tangent and the positive semi-axes $Ox$ and $Oy$ is least?

A sheet of cardboard has the shape of a rectangle with sides $a$ and $b$. By cutting squares from the corners of this rectangle and folding up the protruding parts of the cross-shaped figure, we obtain a box open at the top, whose height equals the side of the square. What should the side of the square be so that the volume of the box is greatest?

A trough is to be built from three boards of equal width. At what angle of inclination of the side walls will the cross-sectional area of the trough be greatest?

Find the height of the regular triangular prism of greatest volume inscribed in a sphere of radius $R$.

A circle of radius $R$ is divided into two segments by a line $l$ at distance $h$ from the center of the circle. Among all rectangles inscribed in the smaller of these segments, find the rectangle with the greatest area.

Find the greatest volume of a cylinder whose axial cross-section has perimeter $a$.

Compute the greatest volume of a cylinder whose total surface area equals $S$.

A tin can has a cylindrical shape. Find the most advantageous dimensions of the can, i.e., determine the ratio of the diameter of the base to the height of the cylinder having the greatest volume for a given total surface area.

What should a vessel be, consisting of a cylinder topped with hemispheres, with walls of given thickness, so that for a given capacity $v$ the least amount of material is used on it?

Determine the ratio of the radius of the base to the height of the cylinder having the least total surface area for a given volume.

Find the greatest total surface area of a cylinder inscribed in a sphere of radius $R$.

Among all cylinders inscribed in a cube with edge $a$ such that the axis of each cylinder coincides with the diagonal of the cube, and the circles of the bases are tangent to the faces of the cube, find the cylinder of greatest volume.

Find the height of the cone of greatest volume inscribed in a sphere of radius $R$.

Find the height of the cone of least volume circumscribed about a sphere of radius $R$.

A cylinder of greatest volume is inscribed in a cone whose base radius equals $R$ and height $H$. Find the base radius and height of this cylinder.

A sector is cut from a circular sheet of tin and rolled into a conical funnel. What should the angle of the sector be so that the funnel has the greatest volume?

Find the least lateral surface area of a cone having volume $v$.

Find the greatest volume of a cone with given slant height $l$.

Find the least volume of a cone circumscribed about a hemisphere of radius $r$ (it is assumed that the bases of the hemisphere and the cone lie in the same plane and are concentric).

Consider the pencil of lines passing through the point $M(a ; b)$, where $a > 0, b > 0$, and intersecting the positive semi-axes $Ox$ and $Oy$. Find the least length of the segment $PQ$, where $P$ and $Q$ are the points of intersection of a line of the pencil with the positive semi-axes.

A stone is thrown with a given initial speed at an angle $\alpha$ to the horizontal. Neglecting air resistance, determine for what $\alpha$ the range of the stone's flight will be greatest.

The internal resistance of a galvanic cell equals $r$. At what external resistance will the power of the current obtained from this cell in the external circuit be greatest?

A uniform rod of length $l$, where $2 R < l \leqslant 4 R$, is placed in a cup shaped like a hemisphere of radius $R$. Find the equilibrium position of the rod.

A canal of width $b$ is built at a right angle to a river of width $a$. Find the greatest length of a log that can be floated from the river into this canal.

To reduce the friction of the liquid against the walls of a canal, the area wetted by the water should be as small as possible. Show that the best shape of an open rectangular canal with a given cross-sectional area is the one in which the width of the canal is twice its height.

A beam with a rectangular cross-section is hewn from a round log. Assuming that the strength of the beam is proportional to $ah^{2}$, where $a$ is the base and $h$ the height of the rectangle, find the ratio $h / a$ for which the beam has the greatest strength.

A vessel with a vertical wall of height $h$ stands on a horizontal plane. A jet spurts from a hole in the wall of the vessel. Determine the position of the hole for which the range of the jet is greatest, if the speed of the outflowing liquid equals $\sqrt{2 g x}$, where $x$ is the depth of the hole (Torricelli's law).

A factory $A$ is to be connected by a highway to a straight railway on which a settlement $B$ is located. The distance $AC$ from the factory to the railway equals $a$, and the distance $BC$ along the railway equals $b$. The cost of shipping goods by highway is $k$ times $(k > 1)$ higher than the cost of shipping by railway. To which point $D$ of the segment $BC$ should the highway be built from the factory so that the cost of shipping goods from the factory $A$ to the settlement $B$ is least?

At what height above the center of a round table of radius $R$ should an electric light bulb be placed so that the illumination of the edge of the table is greatest?

A luminous point is located on the line of centers of two spheres and lies outside these spheres. At what position of the luminous point will the sum of the areas of the illuminated parts of the surfaces of the spheres be greatest?

A load lying on a horizontal plane $P$ is to be moved from its place by a force applied to this load. Determine the angle formed by this force with the plane $P$ for which the magnitude of the force is least, if the coefficient of friction of the load equals $k$.

An observer stands facing a picture mounted on a vertical wall. The lower edge of the picture is above the observer's eye level by $a$, the upper edge by $b$. At what distance from the wall should the observer stand so that the angle at which he sees the picture is greatest?

The point $A\left(x_{0} ; y_{0}\right)$ is located inside the parabola $y^{2} = 2 p x$, $p > 0$, the point $B$ is on the parabola itself, and $F$ is its focus. Prove that the length of the broken line $ABF$ is least if the segment $AB$ is parallel to the axis of the parabola, and the angle $FBA$ is bisected by the normal to the parabola at the point $B$ (the principle of the parabolic mirror).

The points $A$ and $B$ are located respectively in the upper and lower half-planes of the rectangular system $xOy$. A particle moves along the broken line $AMB$, where $M$ is a point of the axis $Ox$. The speeds of the particle's motion in the upper and lower half-planes are respectively $v_{1}$ and $v_{2}$. Prove that the time of motion of the particle is least if $\sin \alpha / \sin \beta = v_{1} / v_{2}$, where $\alpha , \beta$ are the angles formed by the segments $AM$ and $BM$ with the normal to the axis $Ox$ (in optics these angles are called respectively the angle of incidence and the angle of reflection).