5. Find the area of a park 396 ft. by 396 ft. Give your answer in acres.

6. How many acres are in a rectangular farm 1.5 mi. long by 14 mi. wide? Find the value of the farm at $49

an acre.

A township is a tract of land 6 mi. square, and it is divided into 36 sections each 1 mi. square. The sections are numbered as in Fig. 1.

A section is subdivided as indicated in Fig. 2. There are two divisions, E. and W. 1. The E. is divided into two equal squares called N.E. and S.E. . The 4. W. is divided into two equal squares called N.W. and S. W. 1. These are again subdivided as shown.

EXERCISE 64

? in S.E.

1. Draw a figure and locate S. W. of S.E. ; N. of S.E.; N.E. of S.W.;

W.

2. Locate N.E. of the N.E. How many acres are in N.W. acres are in N. W.

S.E.

of S.

? in S. of S. W. †?

3. A man buys

of the S.

1

N.W. of S. W.

Find the

of N. W. and also of the 124

at the rate of $20 an acre.

cost of his purchase.

VOLUMES OF RECTANGULAR SOLIDS

The volume of a rectangular solid is obtained by taking the product of its three dimensions expressed in units of the same denomination.

EXERCISE 65

1. Find the volume of a box 12 ft. by 7 ft. by 6 ft.

2. Find the cubical contents of a room 18 ft. by 12 ft. and 9 ft. high.

3. Find the number of cubic

feet in a room 32 ft. by 24 ft. and 12 ft. high.

4. How many cubic yards of earth must be removed. for the foundation of a house 75 ft. by 54 ft., if the earth to the depth of 21 ft. is removed?

5. A cistern, in the shape of a rectangular solid, is 22 ft. by 14 ft. and 6 ft. deep. How many gallons of water does it contain?

6. A bin is 8 ft. by 3 ft. and 6 ft. high. How many bushels does it hold?

7. In order to build a concrete wall, earth is removed to the depth of 6 ft. If the wall is 210 ft. long and 12 ft. wide, how many cubic yards of earth must be removed?

8. How many cubic yards of gravel are required to fill, to the depth of 6 in., a street 1 mi. long and 36 ft. wide ?

9. How many cubical boxes 2 ft. each way would a storeroom 18 ft. by 12 ft. and 10 ft. high hold?

10. The Sault Ste. Marie Canal is 1.6 mi. long, 160 ft. wide, and 25 ft. deep. Express in cubic yards the volume of water required to fill it.

11. A block of marble is 4 ft. by 3 ft. and 24 ft. long. How many tons does it weigh, if a cubic foot of marble weighs 170 lb. ?

12. How many pounds does a cedar beam 14 in. by 10 in. and 40 ft. long weigh, if a cubic foot of cedar wood weighs 38.1 lb. ?

13. A cubic foot of clay weighs 75 lb. Find, in tons, the weight of a clay bank 10 ft. by 4 ft. and 80 ft. long.

14. A box 9 in. by 8 in. and 6 in. deep is filled with mercury. Find its weight in pounds if a cubic foot of mercury weighs 13,570 oz.

15. How many 3-in. cubes are required to fill a cubical box each of whose edges is 1 yd.?

16. A pile of 4-ft. wood 8 ft. long and 4 ft. high contains a cord. How many cords of wood are in a pile of 4-ft. wood 120 ft. long and 12 ft. high?

17. Find the weight of the water covering an acre to the depth of 4 inches. 1 cu. ft. of water weighs 1000 oz.

A triangle is a portion of a plane bounded by three straight lines. ABC is a triangle.

Two straight lines are parallel if they can never meet no matter how far they may be produced.

A quadrilateral is a portion of a plane bounded by four straight lines. Figure 2 represents a quadrilateral.

A quadrilateral having its opposite sides parallel is called P a parallelogram. Figure 3 is a parallelogram.

The sides AB, CD are parallel. Also the sides AD, BC are parallel.

Consider next the parallelogram ABCD M D

(Fig. 4) and the rec

A

FIG. 1.

S

B

R

FIG. 2.

C

B

FIG. 3.

K

C

AMD from the figure ABCM, the parallelogram remains. If the triangle BKC is taken from the figure ABCM, the rectangle remains. Hence, the rectangle equals the parallelogram in area. But the area of the rectangle is obtained by taking the product of its two dimensions, i.e. AB and BK. Therefore, the area of the parallelogram is equal to AB × BK. AB is called the base of the parallelogram; BK, i.e. the distance between the parallel sides, is called the altitude, or height, of the parallelogram.

Consequently, the area of a parallelogram equals the product of its base by its altitude.

A quadrilateral having two sides parallel is called a trapezoid. ABCD (Fig. 5) is a trapezoid, having AB

parallel to DC. AB and CD are respec

tively the lower and upper bases of the trapezoid.

Take a piece of paper and make a trapezoid. Make another trapezoid just equal to it.

Place them as shown in Fig. 6. You then

have a parallelogram KBCL. Its area is equal to (KA + AB) × height of the parallelogram, i.e. equal to (DC+ AB) × height of the parallelogram.

The trapezoid is 1⁄2 of KBCL.

.. the area of the trapezoid ABCD equals of (DC+ AB) × the height of the trapezoid.

Consequently, the area of a trapezoid equals one half the sum of its parallel sides multiplied by the distance between them.

If one of the bases, e.g. DC, of a trapezoid were to become smaller and smaller, the fig

ure would ultimately be a triangle. Hence, the area of a triangle equals one half of its base multiplied by its height.

This may also be seen readily from Fig. 7.

FIG. 7.

B