Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross Set 8 By Herbert I. Gross and Richard A. Medeiros next.

Answer: 9.0 × 10 8 Solution 1a: The number c × 10 n is said to be written in scientific notation if n is an integer and c is at least 1 but less than 10 (in more mathematical terms, 1< c < 10). next © 2007 Herbert I. Gross In order for the sequence of digits 900,000,000 to represent a number between 1 and 10 the decimal point would have to be between the 9 and the first 0; that is 9.00000000. next

Solution 1a: To “convert” 9.00000000 into 900,000,000 we would have to move the decimal point 8 places to the right; and every time we move the decimal point 1 place to the right we have multiplied the number by 10. next © 2007 Herbert I. Gross next 9 0 0 0 0 0 0 0 0.........,, 1 2 3 4567 8

Solution 1a: Hence, moving the decimal point 8 places to the right means that we have multiplied the number by 8 factors of 10; that is, by 10 8. next © 2007 Herbert I. Gross Therefore, written in scientific notation, 900,000,000 = 9 × 10 8. next

If we wanted to derive the above result using only the properties of exponent arithmetic, recall that 10 -8 × 10 8 = 10 0 = 1. Hence… next Note 1a 900,000,000 = (9,00,000,000 × 10 -8 ) × 10 8 next © 2007 Herbert I. Gross 900,000,000 = 9,00,000,000 × (10 -8 × 10 8 ) By the associative property of multiplication we may rewrite it as… next

Multiplying by 10 -8 is the same as dividing by the reciprocal of 10 -8 ; that is, by 10 8. When a number is written in place value notation, we may divide it by 10 by moving the decimal point 1 place to the left. Since dividing by 10 8 means dividing the number by 10 eight times, we divide a number by 10 8 by moving the decimal point 8 places to the left. next Note 1a next © 2007 Herbert I. Gross next

In other words we may rewrite the expression… next Note 1a = ( 9 0 0 0 0 0 0 0 0 ÷ 10 8 ) next © 2007 Herbert I. Gross 900,000,000 = (9,00,000,000 ÷ 10 8 ) × 10 8 next as… 1 234 5 6 78 7 6 5 4 3 2 1 0 × 10 8

The above note can be summarized less formally by observing that to convert the number 900,000,000 to the single digit 9, we have to move the decimal point 8 places to the left. However, to keep the value of the number the same we “undo” moving the decimal point 8 places to the left by moving it back 8 places to the right. next Note 1a © 2007 Herbert I. Gross

In summary… -- Start with 900,000,000. -- Move the decimal point 8 places to the left to obtain 9.00000000 or 9. -- Move the decimal point 8 places to the right, which is the same as multiplying by 10 8 to obtain 9 × 10 8. next Note 1a © 2007 Herbert I. Gross

9 × 10 8, 90 × 10 7, 900 × 10 6, 9,000 × 10 5, 90,000 × 10 4, etc. are all correct ways to express 900,000,000, but only 9 × 10 8 is correct if we require that 900,000,000 is to be represented in scientific notation. next Note 1a © 2007 Herbert I. Gross 10 8 = 100,000,000 = 1 × 10 8 10 9 = 1,000,000,000 = 1 × 10 9 Hence, 9 × 10 8 is between 100,000,000 and 1,000,000,000.

More generally a number is at least 100,000,000 but less than 1,000,000,000 if and only if it can be written in the form c × 10 8 ‚ where c is at least 1 but less than 10; that is 1≤ c < 10. next Note 1a © 2007 Herbert I. Gross

If we were to represent 900,000,000 in the form, say, 900 × 10 6 rather than in scientific notation, it would obscure the fact that its order of magnitude is 8. next Note 1a © 2007 Herbert I. Gross We say that the order of magnitude of a number is 8 if it can be written in the form c × 10 8 where 1 ≤ c < 10. In other words, to be of magnitude 8, a number has to be at least 10 8 but less than 10 9. More generally, we say that the order of magnitude of a number is n if it can be written in the form c × 10 n where 1 ≤ c < 10 and n is an integer. next

Answer: 7.82003 × 10 5 Solution 1b: There is often a tendency in looking at the notation c × 10 n where 1≤ c < 10 and n is an integer to think that c has to be a whole number. The point of this part of the problem is to help emphasize that the only requirement is that 1≤ c < 10. Thus such numbers as 3.56, √2, 7.82003 etc. are permissible values for c. © 2007 Herbert I. Gross

next Solution 1b: So once we notice that we would have to move the decimal point 5 places to the left in order to convert 782,003 into 7.82003 (that is, a number between 1 and 10) we simply move the decimal point 5 places to the left and then “undo” this by multiplying by 10 5. © 2007 Herbert I. Gross next × 10 5 7.82003 In summary… 782,003 = (782,003 × 10 -5 ) × 10 5

Solution 1c: In order to convert 0.000000567 into a number between 1 and 10, the decimal point would have to be moved to between the 5 and the 6. To do this we would have to move the decimal point 7 places to the right; that is, we would have to multiply by 10 seven times or 10 7. And to compensate for doing this, we would have to “undo” it by moving the decimal point back 7 places to the left. That is, we would have to divide by 10 seven times. © 2007 Herbert I. Gross

next Solution 1c: This verbal explanation can be summarized more succinctly by the following sequence of steps… © 2007 Herbert I. Gross 5.67 × 10 -7 (0.000000567 × 10 7 ) × 10 -7 = 0.000000567 × (10 7 × 10 -7 ) =0.000000567 =

next Note 1c © 2007 Herbert I. Gross Order of magnitude is a good way of getting a quick estimate of the size of a number. For example, if the number is 5.67 × 10 11, we see immediately that is greater than 10 11 but less than 10 12. In other words since 10 11 = 100,000,000,000 and since 100,000,000,000 has 12 digits, any number whose order of magnitude is 11 has 12 digits to the left of the decimal point.

The same logic applies to orders of magnitude that are negative. For example by our definition, the order of magnitude of 5.67 × 10 -7 is - 7. Since 5.67 is greater than 1, 5.67 × 10 -7 is greater than 1 × 10 -7. next Note 1c © 2007 Herbert I. Gross It is also much less cumbersome to write 5.67 × 10 11 than it is to write 567,000,000,000. next On the other hand, 5.67 is less than 10 which means that 5.67 × 10 -7 is less than 10 × 10 -7 (= 10 1 × 10 -7 = 10 1 + -7 = 10 -6 ).

Note 1c © 2007 Herbert I. Gross Hence, the fact that the order of magnitude of 5.67 × 10 -7 is -7 means that it is between 10 -7 ( = 0.0000001) and 10 -6 ( = 0.000001). And since 0.0000001 has six 0's to the right of the decimal point, any number whose order of magnitude is -7 is less than 1, and its decimal representation starts with six and only six 0's to the right of the decimal point.

In summary, the definition of 10 -n that 10 n × 10 -n = 1 (or equivalently that 10 -n × 10 n = 1) makes it relatively easy to rewrite a given number in scientific notation. next Note 1c © 2007 Herbert I. Gross And, just as in the case of positive orders of magnitude, it is much less cumbersome, for example, to write 8.34 × 10 -15 than it is to write (and read) 0.000000000000000834. next

Answer: 1.08 × 10 6 Solution 2: By changing the order and the grouping of the factors we may rewrite… © 2007 Herbert I. Gross in the equivalent form… Since 12 × 9 = 108 and 10 13 × 10 -9 = 10 4, we may rewrite it in the equivalent form… 108 × 10 4 (12 × 9) × (10 13 × 10 -9 )12 × 10 13 × 9 × 10 -9 next

Solution 2: The problem is that 108 × 10 4 is not in scientific notation. So we then rewrite 108 as 1.08 × 10 2, whereupon… © 2007 Herbert I. Gross becomes… 1.08 × 10 6 next 1.08 × (10 2 × 10 4 ) (1.08 × 10 2 ) × 10 4 108 × 10 4 And since 1.08 is greater than 1 but less than 10, 1.08 ×10 6 is in scientific notation,

Note 2 © 2007 Herbert I. Gross By way of review, in converting 108 × 10 4 into 1.08 × 10 6, the formal procedure is based on the fact that in order to convert 108 into a number that was between 1 and 10, we would have to move the decimal point 2 places to the left (that is, we would have to multiply by 10 -2 ), but in order not to change the value of the number, we would then have to multiply by 10 2 to move the decimal point back 2 places to the right.

next Note 2 © 2007 Herbert I. Gross Stated in more mathematical terms we multiplied 108 by 1 where 1 was written in the form 10 -2 × 10 2. In summary… 1.08 × 10 2 (108 × 10 -2 ) × 10 2 = 108 × (10 -2 × 10 2 ) = 108 =

next Note 2 © 2007 Herbert I. Gross As an example of the above note, suppose we wanted to write 234,854, 409, 596 in scientific notation. We see that to convert this number into a number that was between 1 and 10 we would have to move the decimal point 11 places to the left.

Answer: 9 × 10 6 Solution 3: In problems such as this it is unnecessary to use scientific notation until we get to the final answer. Sometimes it is better to work with whole numbers rather than with decimal fractions. With this in mind we may rewrite 180,000 as 18 × 10 4, 0.00007 as 7 × 10 -5, and 0.0000014 as 14 × 10 -7. © 2007 Herbert I. Gross

Solution 3: If we now substitute these results into… © 2007 Herbert I. Gross which is already in scientific notation. next 910 6 18 × 7 14 × 10 4 × 10 -5 × 10 7 × 18 × 7 14 10 4 × 10 -5 × 10 7 × we obtain...

Answer: 121.287375m 3 Solution 4a: Recall that if s represents the length of the side of a cube, then s 3 represents the volume of the cube. © 2007 Herbert I. Gross Since the measurement is accurate to the nearest tenth of a meter, the length of the side must be between 4.9 meters and 5.1 meters but closer in value to 5.0 meters. next

Solution 4a: If we let s denote the length of a side of the cube (measured in meters) then… © 2007 Herbert I. Gross 4.95 < s < 5.05 And since the volume (V) of the cube is s 3 we see that… 4.95 3 < s 3 < 5.05 3 next We see that s 3 > 4.95 3 and since 4.95 3 = 121.287375, we may conclude that the volume of the cube must be greater than 121.287375 cubic meters.

Answer: 128.787625m 3 Solution 4b: This is simply a continuation of part (a). © 2007 Herbert I. Gross More specifically we see from the equation 4.95 3 < s 3 < 5.05 above that s < 5.05 3, and since 5.05 3 = 128.287375, we may conclude that the volume of the cube must be less than 128.287375 cubic meters. next

Note 4 © 2007 Herbert I. Gross In the language of scientific notation, we would write 5.0 × 10 0, or more simply 5.0 to indicate that the measurement was accurate to the nearest tenth. In pure mathematics 5, 5.0, and 5.00 mean the same thing. However in the world of real life measurements, 5.0 indicates that the measurement was accurate to the nearest tenth while 5.00 indicates that the measurement was accurate to the nearest hundredth.

next Note 4 © 2007 Herbert I. Gross In a typical textbook math problem one might see a problem such as “If the length of each side of a cube is 5 meters, what is the volume of the cube?”. And the correct answer would be “125 cubic meters” However, what we showed in parts (a) and (b) is that if the measurement of the length of a side was accurate to no greater than the nearest tenth of a meter, then the exact volume could be any value between 121.287375 and 128.287375 cubic meters.

next Note 4 © 2007 Herbert I. Gross Of course as the accuracy of the measurement improves the range of possible values for the volume of the cube narrows. For example, suppose that measured to the nearest hundredth of a meter the length of each side was 5 meters. Then the range of values for length in meters of each side of the cube is given by 4.995 < s < 5.005. Hence the volume, in cubic meters, must be between 4.995 3 and 5.005 3.

next Note 4 © 2007 Herbert I. Gross The fact that… 4.995 3 = 124.625374875 and 5.005 3 = 125.375375125 …tells us that the volume of the cube is between 124.625374875 and 125.375375125 cubic meters. In particular to the nearest cubic meter, the volume is 125 cubic meters. next

Note 4 © 2007 Herbert I. Gross To see why measured to the nearest hundredth, the length of each side is between 4.995 and 5.005 meters, think of 5 written in the form 500 hundredths. If we're counting by hundredths, 500 hundredths comes after 499 but before 501 hundredths.

next Note 4 © 2007 Herbert I. Gross So to be closest to 500 hundredths, the measurement has to be greater than 499.5 hundredths but less than 500.5 hundredths. In other words, the length has to be between 4,995 and 5,005 thousandths; and in the language of decimals this means the length is between 4.995 and 5.005 meters.

next To the nearest millimeter (that is, to the nearest tenth of a centimeter), the length of a rectangle is 7.4 cm; and to the nearest centimeter the width of the rectangle is 5 cm. What is the least area the rectangle can have? Problem #5a © 2007 Herbert I. Gross Answer: 33.075cm 2 next

Answer: 33.075cm 2 Solution 5a: The fact that to the nearest millimeter, the length (L) of the rectangle is 7.4 centimeters means that… © 2007 Herbert I. Gross 7.35 < L < 7.45 next And the fact that to the nearest centimeter the width (W) is 5 centimeters means that… 4.5 < W < 5.5

Solution 5a: The area (A) of a rectangle is the product of its length (L) and width (W). From 7.35 4.5 and L > 7.35... Hence the area (A = L × W) must be greater than 7.35 × 4.5 ( = 33.075) square centimeters. © 2007 Herbert I. Gross

next To the nearest millimeter (that is, to the nearest tenth of a centimeter), the length of a rectangle is 7.4 cm; and to the nearest centimeter the width of the rectangle is 5 cm. What is the greatest area the rectangle can have? Problem #5b © 2007 Herbert I. Gross Answer: 40.975cm 2 next

Answer: 40.975cm 2 Solution 5b: This is a continuation of the previous part where we saw that L < 7.45 and W < 5.5. © 2007 Herbert I. Gross next Hence the area (A = L × W) must be less than 7.45 × 5.5 ( = 40.975) square centimeters.

© 2007 Herbert I. Gross In the “old days”, before the advent of the hand held calculators, significant figures were used to make approximations that simplified calculations. For example, in the same vein that a chain is no stronger than its weakest link, when we work with significant figures a computation can not be any “stronger” than that of its least accurate measurement. Concluding Notes On Significant Figures

next © 2007 Herbert I. Gross So for example, suppose we wanted to find the circumference of a circle whose diameter, measured accurately to the nearest inch, was 20 inches. The formula for the circumference (C) of a circle whose diameter is D is given by C = π D. The number π, to five decimal digit accuracy is 3.14159 Concluding Notes On Significant Figures

next © 2007 Herbert I. Gross Since the measurement of the diameter is only accurate to the nearest inch, we know that it must be between 19.5 and 20.5 inches. Hence if we take the value of π to be 3.14159, we see that the circumference of the circle is at least 3.1459 × 19.5 inches, but less than 3.14159 × 20.5 inches. Performing the calculations we see that… 61.261005 < C < 64.402595 Concluding Notes On Significant Figures

next © 2007 Herbert I. Gross We see from 61.261005 < C < 64.402595 that measured to the nearest 10 inches, the circumference of the circle is 60 inches. Other than for that the exact value is somewhat in doubt. Concluding Notes On Significant Figures

next © 2007 Herbert I. Gross One might expect that using 3.14159 for the value of π would give us a more accurate estimate than if we use 3.14 for the value of π. However, 3.14 × 19.5 = 61.23 and 3.14 × 20.5 = 64.37. Hence even with this weaker approximation, the circumference has to be between 61.23 and 64.37 inches; and to the nearest 10 inches the answer is still 60 inches. Concluding Notes On Significant Figures

next © 2007 Herbert I. Gross If fact if we even estimated π by letting it equal 3, the circumference, measured to the nearest 10 inches, still would have been 60 inches. The point is that without a calculator it is easier to multiply a number by 3 or 3.14 than by 3.14159; and that's why so many rules of thumb existed for how to round off numbers. Concluding Notes On Significant Figures next

Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross Set 8 By Herbert I. Gross and Richard A. Medeiros next.

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Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross Set 8 By Herbert I. Gross and Richard A. Medeiros next.

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