Math 2000 Solutions 5

Spring 2002

Unit 3B

14. We need to write 2,626,165 gigawatt-hours in scientific notation. First note that 2,626,165 = 2.626165 × 10⁶ and that 1 gigawatt-hour = 1 billion watt-hours = 10⁹ watt-hours. Therefore,

2,626,165 gigawatt-hours = 2.626165 × 10⁶ × 10⁹ watt-hours = 2.626165 × 10¹⁵ watt-hours

(adding exponents).

28. As said in class, this problem relies on estimates. Let’s suppose that there are 100 million = 10⁸ cars in the United States (actually, in 2001, there were about 130 million registered vehicles). Let also suppose that on average each car gets filled with 15 gallons of gasoline once a week. This means that the average car consumes 15 gallons/week × 52 weeks/year = 780 gallons/year, which we can estimate at 800 gallons per year. Finally, we are told that each gallon of gasoline produces10.2 kilograms of carbon dioxide (CO₂) when burned. So the total amount of carbon dioxide produced in a year is

A rough estimate is that in one year, 800 billion kilograms of CO₂ are produced by cars in this country.

30. Please use units to make this conversion!

There are about 82 suicides per day in this country.

48. The conversion factor for this time line is 15 billion years = 100 meters, or 1.5 × 10¹⁰ years = 10² meters. To convert 1 billion years to meters, we use the conversion factor:

Thus, the point in time 1 billion years ago is 6.7 meters (about 7 yards) from the end of the field that represents the present. Converting 10,000 (10⁴ years), we have

The point in time 10,000 years ago is 0.0000667 meters (or 0.067 millimeters) from the end of the field that represents the present.

Percentage Problems

3a,b. These two problems can be worked together because they use the same sentence. As discussed in class, the sentence that describes salary before and after income tax is

After-tax salary = Before-tax salary - (tax rate) × before-tax salary.

In other words, income tax is subtracted from before-tax salary, not added to after-tax salary. In exercise 3a, we have a 28% tax rate, so the sentence is

After-tax salary = Before-tax salary - (28% × before-tax salary).

Noting that before-tax salary means 100% of before-tax salary, we have

After-tax salary = (100% ×Before-tax salary) - (28% × before-tax salary)

= 72% × before-tax salary.

For exercise 3a, we are given that after-tax salary is $44,500, so we have

$44,500 = 72% × before-tax salary = 0.72 × before-tax salary.

To find before-tax salary, we divide both sides of the sentence by 0.8 and find that

The after-tax salary is $61,806. You should check to see if this is correct: if you subtract 28% of $61,806 from $61,806, you get the after-tax salary $44,500.

For exercise 3b, we use the same sentence:

After-tax salary = (100% × before-tax salary) - (28% × before-tax salary).

= 72% × before-tax salary.

Now we are given that the before-tax salary is $52,500. The after-tax salary can be found directly by multiplication:

After-tax salary = 0.72 × $52,500 = $37,800.

The after-tax salary is $37,800.

3c. This is a comparison problem and we must find the relative difference. The wording of the sentence tells us that 1 kilometer is the reference quantity (because it follows than) and 1 mile is the compared quantity. We have

Noting that 1 mile = 1.61 km, we can write

Thus, 1 mile is 61% greater than 1 kilometer.

3d. Again, the sentence is critical. Because the stock market increased, the opening and closing values of the stock market are related by the sentence

Closing value = (100% of opening value) + (4.5% of opening value) = 104.5% × opening.

We are given that the closing value is 8664. Thus,

8664 = 104.5% × opening = 1.045 × opening.

To find the opening value, we divide by 1.045 to find

The market opened at 8291. You should check that this is correct: A 4.5% increase to 8291 gives 8664.