Square Root.

314. MISCELLANEOUS PROBLEMS.

1. What is one of the two equal factors of 9216 ?

2. What is one of the four equal factors of 20736? *

3. If 7921 soldiers were arranged in a solid square, how many soldiers would there be on each side?

4. How many rods of fence will enclose a square field whose area is 40 acres?

5. How many rods long is one side of a square piece of land containing exactly one acre? †

6. If the surface of a cubical block is 150 square inches, what is the length of one edge of the cube?

7. How many rods of fence will enclose a square piece of land containing 4 acres 144 square rods?

8. Find the side of a square equal in area to a rectangle that is 15 ft. by 60 ft.

9. Compare the amount of fence required to enclose two fields each containing 10 acres: one field is square, and the other is 50 rods long and rods wide.

10. Find the area of the largest possible rectangle having a perimeter of 40 feet.

11. If a square piece of land is of a square mile, how much fence will be required to enclose it?

* To find one of the four equal factors of a number (the 4th root) extract the square root of the square root. Why? What is the fourth root of 81?

Find the answer to problem 5, true to hundredths of a rod.

Algebra.

315. SQUARE ROOT AND AREA.

1. If a piece of land containing 768 square rods is three times as long as it is wide, how wide is it?*

2. If a certain room is twice as long as it is wide, and the area of the floor 968 square feet, what is the length and the breadth of the room?

3. One half of the length of Mr. Smith's farm is equal to its breadth. The farm contains 80 acres. How many rods of fence will be required to enclose it?

4. Each of four of the faces of a square prism is an oblong whose length is twice its breadth. The area of one of these oblongs is 72 square inches. What is the solid content of the prism.

5. The width of a certain field is to its length as 2 to 3. Its area is 600 square rods. The perimeter of the field is how many rods?

6. If & of the length of an oblong equals the width and its area is 768 square inches, what is the length of the oblong? times the square of a number you add 15 the What is the number?

7. If to 2 sum is 375.

*To solve this problem arithmetically, one must discern that this piece of land can be divided into three equal squares, the side of each square being equal to the width of the piece.

Algebra.

316. SQUARE ROOT AND PROPORTION.

When the same number forms the second and the third term of a proportion it is called a mean proportional, of the first and the fourth term; thus, in the proportion 3:6::6:12, 6 is a mean proportional of 3 and 12.

EXAMPLE.

In the proportion 12:x:: x: 75, find the value of x. Since the product of the means equals the product of the extremes, x times x equals 12 times 75, or,

Find the value of x in each of the following proportions:

(a) Find the sum of the six mean proportionals.

7. An estate was to be divided so that the ratio of A's part to B's would equal the ratio of B's part to C's. If A received $8000 and C received $18000, how much should B receive?

8. Find the mean proportional of

9. The ratio of the areas of two

What is the ratio of their lengths?

and 13.

squares is as 4 to 9.

10. The area of the face of one cube is to the area of the

face of another cube as 16 to 25.

solid contents of the cubes?

What is the ratio of the

here given, and by careful measurements and paper cutting, that the square of the hypothenuse of a right-triangle is equivalent to the sum of the squares of the other two sides.

Figures 2 and 3 are equal squares. If from figure 2, the four right-triangles, 1, 2, 3, 4, be taken, H, the square of the hypothenuse, remains. If from figure 3, the four right-triangles (equal to the four right-triangles in figure 2) be taken, B, the square of the base, and P, the square of the perpendicular, remain. When equals are taken from equals the remainders are equal, therefore the square, H, equals the sum of the squares B and P.

3. To find the hypothenuse of a right-triangle when the base and perpendicular are given: Square the base; square the perpendicular; extract the square root of the sum of these squares.

318. Miscellaneous Review.

1. Find approximately the diagonal of a square whose side is 20 feet.*

2. Find approximately the distance diagonally across a rectangular floor, the length of the floor being 30 feet and its breadth 20 feet.

3. How long a ladder is required to reach to a window 25 feet high if the foot of the ladder is 6 feet from the building and the ground about the building level?

4. If the length of a rectangle is a, and its breadth b, what is the diagonal?

5. The base of a right triangle is 40 rods and its perpendicular, 60 rods. (a) What is its hypothenuse? (b) What is its area? (c) What is its perimeter ?

6. The area of a certain square piece of land is 21 acres. (a) Find (in rods) its side. (b) Find its perimeter. (c) Find its diagonal, true to tenths of a rod.

7. The length of a rectangular piece of land is to its breadth as 4 to 3. Its area is 30 acres. (a) Find its breadth. (b) Find its perimeter. (c) Find the distance diagonally across it.

8. A certain piece of land is in the shape of a right-triangle. Its base is to its altitude as 3 to 4. Its area is 96 square rods. (a) Find the base. (b) Find the altitude. (c) Find the perimeter.

9. Find one of the two equal factors of 93025.

*From the study of right-triangles on page 359 it may be learned that the diagonal of a square is equal to the square root of twice the square of its side.