549. The total surface of a regular tetrahedron is 60 sq. m. Find its volume.

550. Find the locus of those points of a plane at which a given straight line, not lying in that plane, subtends a right angle. Show when the locus becomes a point and disappears.

551. The volume of a sphere inscribed in a regular tetrahedron is 2 cu. in. What is the volume of a circumscribed sphere?

552. Show how to cut a given polyhedral angle of four faces so that the section shall be a parallelogram.

553. What part of the surface of a sphere is illumined by a lamp placed at a distance of a diameter from the surface of the sphere?

554. Two tetrahedrons which have a trihedral angle of one equal to a trihedral angle of the other are to each other as the products of the three edges of the equal trihedral angles.

555. A right circular cone the altitude of which is three times the radius of its base, and a sphere, the radius of which is equal to the radius of the base of the cone, are immersed in a rectangular cistern of water the base of which is 9 ft. by 11 ft. If they are removed, the water level is lowered by 2 ft. Find their dimensions.

556. A cylindrical tank 10 ft. long and 5 ft. in diameter lies with its axis horizontal. If it contains gasoline to a depth of 15 in., how many gallons are in the tank? (231 cu. in. to the gallon.)

557. Find the point the great circle distances of which from the sides of a spherical triangle are equal. State your construction.

558. Four of the six planes determined by the diagonals of a parallelepiped divide the parallelepiped into six quadrangular pyramids. Prove that these six pyramids are equivalent.

559. Describe a spherical surface with a given radius that shall pass through two given points and be tangent to a given plane.

560. The axis of a right cylinder passes through the centers of all sections parallel to the base.

561. The base of a pyramid is a right triangle whose base is 12 in. and whose hypotenuse is 20 in. The altitude of the pyramid is 15 in. Find the volume of the frustum of this pyramid cut off by a plane 3 in. above the base.

562. The base of a right prism is a rhombus, one side of which is 10 in., and the shorter diagonal is 12 in. The altitude is 15 in. Find the volume.

563. A regular hexagonal pyramid is volume by a plane parallel to the base.

cut into two parts of equal What is the distance from

the vertex to this plane, in terms of the altitude?

564. Find the volume of a right pyramid whose slant height is 13 ft., and whose base is an equilateral triangle inscribed in a circle whose radius is 10 ft.

565. The volume of the frustum of a square pyramid is 74 cu. in.; the edges of the bases are 3 in. and 4 in. respectively. Find the altitude.

566. The altitude of a regular pyramid is 2 a, and the base is a triangle inscribed in a circle of radius a. Find the lateral area of the pyramid.

567. Find the area of a zone on a sphere of radius r that is illumined by a lamp placed at a distance a from the surface.

568. A cone of wood has its vertex angle equal to 60°, and the radius of its base equal to 2 in. A cylindrical hole of radius 1 in. is bored through the entire cone, the axes of the two coinciding. How much of the cone goes into chips?

569. The distance between two parallel planes is 16 in. A line 24 in. long has an extremity in each of these planes. Find the length of the segments into which this line is divided by a plane parallel to the given planes and 4 in. from one of them.

570. What is the greatest number of faces that a convex polyhedral angle can have if each face angle is 60°? Why?

571. If the area of a section is one third that of the base, what is the ratio of the segments into which the altitude is divided?

572. Define polar triangle of a spherical triangle. If a is the side of a spherical triangle, and A' the opposite angle of the polar triangle, prove that A' + a = 180°.

573. From a point 6 ft. from the surface of a sphere, one quarter of its surface is visible. Find the radius of the sphere.

574. A triangle whose sides are respectively 15 in., 13 in., and 4 in., revolves about the shortest side as an axis. Find the volume of the solid generated by the revolving triangle.

575. The volume of a regular hexagonal prism is 81√3; the altitude of the prism is equal to the longest diagonal of the base. Find the total area of the prism.

576. Find the volume of a hemisphere whose entire surface equals S. 577. Find the locus of a point on a sphere that is equidistant from two given points on the surface.

578. The volume of a sphere is 4500 π cu. in. Find its surface. 579. Find the surface and the volume of the solid generated by a door 3 ft. wide and 8 ft. high swinging in an arc of 144°.

580. The hypotenuse of a right triangle is 5 in., one of its legs is 3 in. Find the volume of the solid generated by revolving the triangle on its hypotenuse as an axis.

581. Assuming that the radius of the earth is 4000 mi. and that the crust is 30 mi. thick, find the volume of the crust of the earth.

582. Prove that if a line is parallel to one plane and perpendicular

to another, the two planes are perpendicular to each other.

583. Find the weight of 52,800 linear feet of copper wire of an inch in diameter. (1 cu. ft. of copper weighs 556 lb.)

584. Find the number of cubic feet of earth in a railway embankment 2500 ft. long, 10 ft. high, 12 ft. wide at the top, and 42 ft. wide at the bottom.

585. Find the cost, at $2.50 a square foot, of gilding a hemispheric dome whose diameter is 50 ft.

586. A sphere of lead 10 in. in diameter is melted and cast into a cone 10 in. high. Find the diameter of the base of the cone.

587. Find the capacity in cubic inches of a berry box in the form of a frustum of a pyramid 5 in. square at the top, and 41⁄2 in. square at the bottom, and 24 in. deep.

588. Two tanks are in form similar solids; one holds 128 gal., the other 250 gal. If the first is 20 in. deep, find the depth of the second.

589. The total surface of a cube is 450 sq. in. Find the volume.

590. Define as loci (1) the intersection of two planes; (2) the bisecting plane of a dihedral angle; (3) the plane which is the perpendicular bisector of a given sect; (4) the plane perpendicular to a given line at a given point.

591. Define (1) the angle between a line and a plane; (2) a parallelepiped; (3) symmetric polyhedral angles; (4) the volume of a geometric solid.

592. Find the number of cubic yards of dirt to be excavated in digging a canal, 50 ft. wide at the top, 30 ft. wide at the bottom, 14 ft. deep on the average, between two locks 2.6 mi. apart.

593. Compute the volume of a regular tetrahedron whose slant height is √3.

594. The sides of a parallelogram, which are 12 in. and 8 in. respectively, form an angle of 60°. Find the volume and the convex surface of the solid generated by the revolution of the parallelogram about one of its longest sides as an axis.

595. Prove that the smallest section of a sphere made by a plane passing through a given point within the sphere is that made by a plane perpendicular to the radius through the given point.

596. The lateral area of a right cylinder is 48 ; the volume is Find the radius and the height of the cylinder.

96 π.

597. An isosceles trapezoid revolves about its longer base as an axis; the bases are respectively 14 in. and 8 in., the legs each 5 in. Find the surface of the solid generated.

598. The radius of a sphere is 20 in. Find the area of a section made by a plane 5.6 in. from the center of the sphere.

599. Find all possible locations of a point that is equidistant from two given points in space and at a given distance from a third point.

600. A cone 5 ft. high is cut by a plane parallel to the base and 2 ft. from the base; the volume of the frustum formed is 294 cu. ft. Find (a) the volume of the cone, (b) the volume of the part cut off by the plane.

601. A cylindrical tank 20 ft. long and 9 ft. in diameter lies with its axis horizontal. If it contains gasoline to a depth of 6 ft., how many gallons are in the tank? (231 cu. in. to the gallon.)

APPENDIX

318. Statements whose Conditions and Conclusions may have Two or More Parts. If the condition or the conclusion of a statement is composed of two or more parts, the consideration of its related statements can be made very complicated by taking those parts in all their different combinations as condition and conclusion. The full discussion of such possibilities has no place in Geometry, but belongs in a course on Logic. The following illustration will, however, show how the different combinations can be used to find new propositions. It will be considered only in reference to the converses.

In an isosceles triangle, a line through the vertex parallel to the base bisects the exterior angles at the vertex.

CONDITIONS:

The triangle is isosceles,

The line passes through the vertex,

The line is parallel to the base.

CONCLUSION:

The line bisects the exterior angles at the vertex. (Since the bisector of one angle also bisects the other, this is considered as one part.)

COMPLETE CONVERSE: The interchange of the entire condition with the entire conclusion would give

If a line bisects the exterior angles at the vertex of a triangle, (1) the triangle is isosceles; (2) the line passes through the vertex; (3) the line is parallel to the base.