Math 120- Quantitative Literacy: 3C Homework

What are significant digits?

Significant digits are the digits in a number that represent actual measurements and therefore have meaning.

How can you tell whether zeros are significant?

The position of zeros in a number with respect to the position of the nonzero numbers in a number is what determines the significance of zeros.

Why can it be misleading to give measurements with more precision than is justified by the measurement process?

It is misleading because the measurement would be perceived as having a greater amount of detail than it actually has

Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.

Next year’s federal deficit will be $675.734 billion.

This statement does not make sense because there are too many significant digits.

Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning.

A $1 million error may sound like a lot, but when compared to our company’s revenue it represents a relative error of only 0.1%.

This statement makes sense because the relative error is low. Thus, the $1 million dollar error is not very big when compared with the actual revenue

Carry out the indicated operation and give your answer with the specified number of significant digits.

41 × 32.6    ; 3 significant digits

1340

Carry out the indicated operation and give your answer with 2 significant digits.

231.89 ÷ 0.034

6800

Describe possible sources of random and systematic errors in the given measurement.

A count of every different meadowlark that visits a three-acre region over a 2-hour period.

Random errors could occur due to not counting some birds and double counting other birds

Find the absolute and relative errors in the following situation.

Your true height is 68.0 inches (5’8"), but a nurse in your doctor’s office measures your height as 68.8 inches.

The absolute error is 0.8 inches.
The relative error is 1.2%

For the pair of measurements, state which one is more accurate and which one is more precise.

Your true height is 62.20 inches. A tape measure that can read to the nearest ½ inch gives your height as 62 ½ inches. A new laser device at the doctor’s office that gives readings to the nearest 0.01 inch gives your height as 62.24 inches.

The laser device is both more accurate and more precise

For the pair of measurements, state which one is more accurate and which one is more precise.

Your weight is 52.26 kilograms. A scale at a health clinic that gives weight measurements to the nearest 1/8 kilogram gives your weight as 53 kilograms. A digital scale at the gym that gives readings to the nearest 0.2 kilogram gives your weight as 52.8 kilograms.

The digital scale is more accurate and the scale at the health clinic is more precise.

Suppose you want to cut 20 identical boards of length 3 feet. The procedure is to measure and cut the first board, then use the first board to measure and cut the second board, then use the second board to measure and cut the third board, and so on. Answer part a and b.

A.) What are the possible lengths of the 20th board, if each time you cut a board, there is a maximum error of ± ½ inch?
B.) What are the possible lengths of the 20th board if, each time you cut a board, there is a maximum error of ± 0.5%?

A.) The board could be as short as 26 inches or as long as 46 inches. 
B.) The board could be as short as 32.57 inches or as long as 39.78 inches.

Round 139.0857 to the nearest whole number.

139

State the number of significant digits and the implied precision of the given number.

1.87345 miles

The number 1.97345 has 6 significant digits.

The number is precise to the nearest hundred-thousandth of a mile.

State the number of significant digits and the implied precision of the given number. 

178.7625 pounds. 

The number 178.7625 has 7 significant digits.

The number is precise to the nearest ten-thousandth of a pound. 

Find the absolute and relative errors in the following situation.

Your homemade pasta sauce recipe calls for 9 ounces of olive oil. You mistakenly use 8 ¼ ounces.

The absolute error is -0.75 ounces.

The relative error is -8.3%

Use the appropriate rounding rules to do the following calculations. Express the result with the correct precision or correct number of significant digits.

Subtract 0.6 pound from 34 pounds to find the weight of your dog before he ate dinner.

33 pounds

Use the appropriate rounding rules to do the following calculations. 
Express the result with the correct precision or correct number of significant digits. 

As you drive down the freeway, a sign tells you that it is a 8 miles to city hall. Your destination lies 3.9 miles beyond city hall. How much farther do you have to drive? 

12 miles

Discuss possible sources of error in the following measurements. Then state whether you think the measurement is believable, given the precision with which it is stated.

The tallest building in the world is Burj Dubai in the United Arab Emirates, with a height of 829.8 meters (2722 feet).

Small random or systematic errors could be present, but the figure is believable with the given precision.

random errors

occur because of random and inherently unpredictable events in the measurement process

systematic errors 

occur when there is a problem in the measurement system that affects all measurements in the same way, such as making them all too low or too high by the same amount 

absolute error

describes how far a measured (or claimed) value lies from the true value

absolute error = measured value - true value

relative error

compares the size of the error to the true value

relative error = [(measured value - true value) / true value] x 100%

accuracy 

describes how closely a measurement approximates a true value; has a small relative error

precision

describes the amount of detail in a measurement

rounding rule for addition or subtraction

round the answer to the same precision as the least precise number in the problem

rounding rule for multiplication or division 

round the answer to the same number of significant digits as the measurement with the fewest significant digits