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Locus in a Plane 7. The locus of the vertex of a right triangle having a given hypotenuse is the circle having the hypotenuse as its diameter.

Locus in Space 7. The locus of the vertex of a right triangle having a given hypotenuse is the sphere having the hypotenuse as its diameter.



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The analytic (or coordinate) geometry of two dimensions can easily be extended to three dimensions. Figure 17-13 shows the two- and three-dimensional axes; the z-axis is perpendicular to both the x- and y-axes. In the figure, arrows indicate the positive directions, and dashed lines indicate the negative axes. Four specific extensions from geometry in a plane to geometry in space follow.

Fig. 17-13





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components of the document, following the logical structures that have been established The physical structure of an XML document is the structure of the actual data and information XML documents share a logical structure, as de ned by the schema or DTD, and then vary dramatically in physical structure based on the data they contain In the preceding example, the logical <greeting>.. </greeting> could contain Hello there or some other greeting as the physical structure Notice the nesting of the elements The <greeting>..</greeting> tag pair is contained within <letter>..</letter> This careful attention to the nesting of elements is part of what makes a well-formed XML document With only the two elements contained in the preceding example, this document is simple and straightforward; however, as documents grow in complexity, the number of elements, nesting, and well-formedness becomes less apparent and easier to misplace.

Coordinate Geometry in a Plane 1. Coordinates: P1 (x1, y1), P2(x2, y2) 2. Midpoint of P1P2: x1 x2 y1 y2 , yM xM 2 2 3. Distance P1P2: d 2(x2 x1)2 (y2 y1)2 4. Equation of a circle having the origin as center and radius r: x2 y2 r2

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Coordinate Geometry in Space 1. Coordinates: P1 (x1, y1, z1), P2 (x2, y2, z2) 2. Midpoint of P1P2: x1 x2 y1 y2 z1 z2 , yM , zM xM 2 2 2 3. Distance P1P2: d 2(x2 x1)2 (y2 y1)2 (z2 z1)2 4. Equation of a sphere having the origin as center and radius r: x2 y2 z2 r2

This will help to identify any inventory shortages resulting from theft or other causes This method, similar to the gross pro t method, is used to estimate the dollar cost of the ending inventory when a physical count cannot be done, such as in the case of re The procedure for determination under this method is as follows: 1 Beginning inventory and purchases must be recorded at both cost and selling prices 2 Total goods available for sale are then computed on both bases 3 Sales for the period are deducted from the goods available for sale at selling price 4 Ending inventory at selling price is the result of step 3 This amount is then converted to ending inventory at cost by multiplying by the appropriate markup ratio The procedure is illustrated in the example below: Cost Step 1.

The area of each face of the cube in Fig. 17-14 is A S e2. The total surface area S of the cube is then 6e2

The areas of the six rectangles that make up the rectangular solid in Fig 17-15 are A A A lw lh for top and bottom faces for front and back faces

The total surface area S of the rectangular solid is then S 2lw 2lh 4 pr2 2prh 2pr(r h)

Fig 17-14

Fig. 17-15

Fig. 17-16

Beginning inventory $280,000 + Net Purchases for period 110,000 Step 2 Goods available for sale $390,000 Step 3 Net sales for period Ending inventory @ selling price Selling Price $400,000 180,000 $580,000 340,000 $240,000.

Fig 17-17

Finding total surface areas of solids Find, to the nearest integer, the total surface area of (a) A cube with an edge of 5 m (Fig. 17-18) (b) A rectangular solid with dimensions of 10 ft, 7 ft, and 42ft (Fig. 17-19) (c) A sphere with a radius of 1.1 cm (Fig. 17-20)

Fig. 17-18

Fig. 17-19

Fig. 17-20

(a) s (b) s (c) s 6e2 2lw 4pr2 6(52) 2lh 150 m2 2wh 2(10)(7) 2(10)(42)

2(7)(42)

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