multiply decimals; we shall now make use of this knowledge in finding the areas and circumferences of circles. The diameter is a straight line drawn through the center of a circle. It is the longest straight line that can be drawn in a circle. To find the circumference of a circle multiply the diameter by 3.1416. (Fig. 4.) Example: Find the circumference of a circle whose diameter is 11 inches. The example is worked as follows: 3.1416 31416 31416 34.5576 There are 34 inches and the decimal .5576 of an inch in the circumference. Remember that we multiply just as though we were multiplying two whole numbers, and then point off as many decimal places in the product as there are decimal places in the multiplier and the number multiplied taken together. Let us next find the circumference of a piston whose diameter is 16.5 inches. 3.1416 16.5 157080 188496 31416 51.83640 A string 51.8 inches long would almost reach around the piston. We point off five decimal places in the product because there are four decimal places in the number multiplied and one decimal place in the multiplier, making five decimal places in all. The radius of a circle is one-half the diameter. To find the area of a circle multiply the radius by itself and this product by 3.1416. Example: Find the area of a piston whose diameter is 16.5 inches. Suppose we have measured to hundredths of an inch. We must keep three decimal figures. The rule for the number of decimal figures to be retained is always keep one more figure than the smallest figure that is accurately known. If we have measured to hundredths of an inch, we must keep the decimal figure in thousandths place. So we multiply as follows: 3.1416 68.062 62832 188496 251328 188496 213.8235792 The answer is 213.82 square inches. We drop all the figures after hundredths place. We may also find the area by squaring the diameter and multiplying by .7854. If the figure in thousandths place had been 5 or more we should have added 1 to the figure in hundredths place and written the answer 213.83. Examples to be solved: 1. Find the circumference of a piston whose diameter is twelve inches.-Answer, 37.6992 inches. Dropping unnecessary figures, we have 37.7 inches. 2. Find the area of the same piston.-Answer, 113.1 square inches. 3. Find the circumference of a piston whose diameter is 14.5 inches.-Answer, 45.55 inches. 4. Find the area of the same piston. For practical purposes we may cut off the last two figures in the square of the radius. Multiplying 52.56 by 3.1416 we get 165.12 square inches. The complete answer is 165.122396, but the last figures are of no value since they are less than one-hundredth of a square inch, so we drop them. 5. What is the area in square feet of the bottom of a circular tank if the diameter is 12 feet? Answer, 113.1 square feet. 6. What is the circumference of the same tank? -Answer, 37.7 feet. 7. The diameter of a piston is 16 inches, the pressure of steam on the piston is 50 pounds on every square inch. What is the total pressure on the piston? First find the number of square inches in the piston. Then multiply this number by 50, which is the pressure per square inch.-Answer, 10053.12 pounds. 8. The diameter of the piston of an hydraulic elevator is 14 inches. The pressure of the water is 40 pounds on every square inch. What is the total lifting force on the elevator? -Answer, 6157.5 pounds. A rough and ready method of finding the area and the circumference of a circle is to use the number 3 instead of 3.1416. This method is good enough if we are not measuring to so fine a point as hundredths of an inch. Example: Find the circumference and the area of a piston whose diameter is 14 inches. Find the circumference of a circle whose diameter is 15 inches. In finding the area of this circle it is better to use decimals. Square 7.5 and multiply by 3.1416. Example to be solved: 9. Find the circumference and area of a circle whose diameter is ten inches.-Answer, Circumference, 31 inches. Area, 78% square inches. Reducing Common Fractions to Decimals and A decimal is a fraction whose denominator is ten, one hundred, one thousand, or some other power of ten. By power of ten we mean a number that you can obtain by multiplying ten by ten and that result by ten, and continue to multiply by ten and never by anything else until you get for a result the number that is given. One hundred is the second power of ten, for you can get 100 by multiplying 10 by 10. Two tens multiplied give 100. One thousand is the third power of 10. One with zeros after it forms a number which is a power of ten. The denominator of a decimal is known by the position of the decimal point. Thus 0.3 means threetenths. We can write it 10 and we have a common fraction. 0.15 means fifteen-hundredths. If we write it 1500 we have changed the decimal to a common fraction. 0.125 changed to a common fraction is 1000. |