math week 3
How would you describe 1 cmequals=100 km as a ratio?
1 10,000,000
The earth is about 12,760 km in diameter and about 150 million kilometers away from the sun. The nearest stars besides the Sun are about 4.3 light-years away (1 light-yearequals=9.5 times 10 Superscript 12 Baseline km9.5×1012 km). At a scale of 1 to 10 billion, the Sun would be about the size of a grapefruit. How big and how far away would the Earth be on this scale? How far would the nearest stars (besides the Sun) be?
1.3 mm
Approximately how far away would the two objects be on this scale?
15 m
The number of miles the average person travels in a car in a year (as either driver or passenger)
3600 miles; assuming the average person travels 10 miles per day
There are approximately 2.32.3 million deaths per year in country A. Express this quantity as deaths per minute.
4.4
Approximately how far would the nearest stars (besides the Sun) be on this scale?
4100 km
Give an example in which the absolute error is small but the relative error is large.
A chemist has 2.9 mg of substance, but a scale measures 2.1 mg. The absolute error is only 0.8 mg, but the relative error is approximately 28%. Next Question
Give an example in which the absolute error is large but the relative error is small.
A company projects sales of $7.30 billion and true sales turn out to be $7.32 billion.
How would you show a scale of 1 cmequals=100 km graphically
A miniruler 1 cm long marked "100 km."
Describe three common ways of expressing the scale of a map or model.
A scale can be described verbally, in words such as "One centimeter represents one kilometer," graphically, such as a marked miniruler on a map, or as a ratio, where one unit is described in terms of the other.
What is an order of magnitude? Explain why such an estimate can be useful even though it may be as much as 10 times too large or too small.
An order of magnitude estimate specifies only a broad range of values, usually within one or two powers of ten. These estimates are useful because they provide an way to easily tell the difference in groups of values, such as populations.
Sampling techniques can be used to estimate physical quantities. To estimate a large quantity, you might measure a representative small sample and find the total quantity by "scaling up." To estimate a small quantity, you might measure several of the small quantities together and "scale down." In the following problem, describe an estimation technique. How many stars are visible in the sky on the clearest, darkest nights in your hometown?
Divide the sky into small sections. Count the amount of stars in one small section of the sky. Multiply the counted number of stars by the number of sections that the sky was divided into.
Give an example of how "percent" (%) and "percentage points" can differ for the same situation. Choose the correct answer below.
If a savings account that previously offered 2% interest now offers 6% interest, then the percent increase is 200% and the percentage point increase is 4.
How are their meanings related?
If the compared value is P% more than the reference value, it is (100plus+P)% of the reference value.
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.
What is the difference between the terms "percent" (%) and "percentage points"?
Percent is used to describe a relative change or difference. Percentage points are used to describe an absolute change or difference.
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.
Briefly describe scientific notation. How is it useful for writing large and small numbers? How is it useful for making approximations?
Scientific notation is a format in which a number is expressed as a number between 1 and 10 multiplied by a power of 10. It is useful for writing large and small numbers because it offers a way to save space and avoid errors in calculations.
What are significant digits? How can you tell whether zeros are significant?
Significant digits are the digits in a number that represent actual measurements and therefore have meaning.
Sampling techniques can be used to estimate physical quantities. To estimate a large quantity, you might measure a representative small sample and find the total quantity by "scaling up." To estimate a small quantity, you might measure several of the small quantities together and "scale down." In the following problem, describe your estimation technique and answer the question. How thick is a penny? a nickel? a dime? a quarter? Would you rather have your height stacked in pennies, nickels, dimes, or quarters?
Stack some coins to a measureable height such as 2 inches, and then divide the height of the stack by the number of coins in the stack. Do this for each type of coin.
Distinguish between absolute and relative difference. Choose the correct answer below.
The absolute difference is the actual numerical difference between the compared value and the reference value. The relative difference describes the size of the absolute difference in comparison to the reference value and can be expressed as a percentage.
Distinguish between the absolute error and the relative error in a measurement. Give an example in which the absolute error is large but the relative error is small and another example in which the absolute error is small but the relative error is large.
The absolute error describes how far a measured (or claimed) value lies from the true value. The relative error compares the size of the absolute error to the true value and is often expressed as a percentage.
George's computer is 11001100% faster than Nancy's.
The claim could be true because if George's computer is 11001100% faster than Nancy's, then it is 1212 times Nancy's computer speed, which makes sense.
By turning off her lights and closing her windows at night, Maria saved 129129% on her monthly energy bill.
The claim could not be true because her new monthly energy bill would be minus−2929% of the previous monthly energy bill. This is impossible since the bill cannot be negative.
Explain the difference between the key words 'of' and 'more than' when dealing with percentages. How are their meanings related?
The key word 'more than' is used to express the relative change between the referenced value and the compared value. The key word 'of' is used to express the ratio of the compared value to the referenced value.
What is the order of magnitude estimate for the number of steps you take in an average day?
The order of magnitude estimate is 3000 to 4000 steps; assuming an average person walks 2 miles per day.
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.
If John earns 20% more than Mary does, then Mary must earn 20% less than John does.
The statement does not make sense because if John earns 20% more than Mary, then Mary must earn approximately 16.7% less than John does.
The rate of return on our fund increased by 50%, to 15%.
The statement makes sense because if the rate of return on our fund increased by 50%, to 15%, then the previous rate is 10%, which makes sense.
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%.
The 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.
In many European countries, the percentage change in population has been negative in recent decades.
The statement makes sense. A percentage change in population can be negative if the data reflects that.
Does "mistakes made in entering or calculating numbers on the tax return" involve random or systematic error? Explain.
This involves a random error because unintentional mistakes are unpredictable and random.
Does "places where the taxpayer reported income dishonestly" involve random or systematic error? Explain.
This involves a systematic error due to underreporting of income.
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.
The table to the right gives size and distance data for the planets at a certain point in time. Calculate the scaled size and distance for each planet using a 1 to 10 billion scale model solar system.
First, find the scaled size of Mercury. To find the scaled size, divide the actual size by the given scale. StartFraction 4880 km Over 10 Superscript 10 EndFraction 4880 km 1010equals=4.9 times 10 Superscript negative 74.9×10−7 km Notice that it is difficult to understand what distance 4.9 times 10 Superscript negative 74.9×10−7 km represents. Convert 4.9 times 10 Superscript negative 74.9×10−7 km to millimeters, rounding to the nearest tenth, for a better representation. 4.9 times 10 Superscript negative 74.9×10−7 kmequals=0.50.5 mm Now find the scaled distance of Mercury to the Sun. Again, divide the actual distance by the given scale. StartFraction 69.9 million km Over 10 Superscript 10 EndFraction 69.9 million km 1010equals=7.0 times 10 Superscript negative 37.0×10−3 km Again, it is difficult to understand what distance 7.0 times 10 Superscript negative 37.0×10−3 km represents. Convert 7.0 times 10 Superscript negative 37.0×10−3 km to meters, rounding to the nearest tenth, for a better representation. 7.0 times 10 Superscript negative 37.0×10−3 kmequals=7.07.0 m mm=6 decimal places m=4 decimal places
