Everyday Math Demystified, 2nd edition

Stan Gibilisco

Explanations for Quiz Answers in Chapter 6

1. Each tile measures 1/4 of a meter square (or 1/4 meter by 1/4 meter), so each edge measures 1/4 of a meter long. Each edge of the dining room floor measures 5 meters long, which equals 20 times the length of a tile's edge. To completely cover the dining room floor, you'll have to lay down an array of 20 by 20 tiles. That's a total of 20 x 20 = 400 tiles. The correct choice is D.

2. In order to calculate the circumference of a circle, we must multiply its radius by 2π, which equals roughly 3.1416 x 2 or 6.2832. If we assume that the earth's orbit constitutes a perfect circle with a radius of 150,000,000 kilometers, then it has a circumference of 150,000,000 x 6.2832 = 942,000,000 kilometers, rounded off to the nearest million kilometers. The correct choice is B.

3. According to the Pythagorean Theorem, the square of the rectangle's diagonal equals the sum of the squares of any two adjacent sides. One of those sides measures 500 feet long, and the either adjacent side measures 1200 feet long. When you square these numbers, you get 500 x 500 = 250,000 and 1200 x 1200 = 1,440,000. Adding the two squared quantities gives you 250,000 + 1,440,000 = 1,690,000. Now you ask, "What number, when squared, equals 1,690,000?" If your calculator has a square root key, you have an easy task; just take the square root of 1,690,000 to get 1300. (If your calculator doesn't have a square root key, you'll have to go through a trial-and-error process.) The diagonal fence will measure 1300 feet long. Each right-triangle-shaped plot of land will therefore have a perimeter of 500 + 1200 + 1300, or 3000 feet. The answer is C.

4. This problem is actually far easier than the problem posed by Question 3. The interior area of a right triangle equals the base length times the height, divided by 2. In this situation, you can let the "base" be the side of the plot that measures 500 feet long, and the "height" be the side that measures 1200 feet. The area of either right-triangle-shaped half will equal 500 x 1200 divided by 2, or 300,000 square feet. The correct choice is C.

5. The perimeter of the whole rectangular land plot equals the sum of the lengths of the sides. That's 500 + 1200 + 500 + 1200 = 3400 feet. The answer is A.

6. The total length of all the fencing in your shared plot, including the rectangular surround and the diagonal dividing the plot in half, will equal the rectangle's perimeter plus the length of the diagonal. You've figured out that the perimeter of the entire rectangle equals 3400 feet, and the length of the diagonal equals 1300 feet. You'll therefore end up with 3400 + 1300 = 4700 feet of fence. The correct choice is B.

7. If we assume that the earth is a perfect sphere with a diameter of 8000 miles, then its radius equals 4000 miles (half the diameter). In order to find the surface area in square miles, we must square the radius (multiply it by itself) and then multiply the result by 4π. The square of the radius equals 16,000,000. If we take the value of π given in the text as 3.1416, then 4π = 12.5664. When we multiply this value by 16,000,000 using a calculator, we get 201,062,400 square miles. That's 201,000,000 square miles to the nearest million. The correct choice is A.

8. As we did in the solution to Question 7, let's assume that the earth constitutes a perfect, smooth geometric sphere with a radius of 4000 miles. If we want to determine the volume in cubic miles, we must cube the radius, then multiply that result by 4π, and finally divide by 3. The cube of the radius (the radius multiplied by itself, then multiplied by itself again) equals 64,000,000,000. When we multiply that quantity by 4π, which we've decided to consider as 12.5664, we get 804,249,600,000. Dividing this huge number by 3 gives us 268,083,200,000 cubic miles. That's 268,000,000,000 cubic miles to the nearest billion (thousand-million). The answer is A.

9. Each sugar cube measures 1/100 of a meter along each edge (height, width, and depth). Each inside edge of the box measures 1 meter long, which equals 100 times the length of a cube's edge. To completely pack the box with sugar cubes, leaving no spaces in between any of the cubes, you'll have to create an array of cubes that measures 100 high, 100 wide, and 100 deep. That's a total of 100 x 100 x 100 = 1,000,000 sugar cubes. The answer is D.

10. Each tiny plastic cube measures 1/1000 of a meter along each edge, so each inside edge of the box is 1000 times the length of a cube's edge. When our robot packs the box with plastic cubes, leaving no spaces whatsoever so that the box ends up filled with solid plastic, it will have created an array 1000 cubes high, 1000 cubes wide, and 1000 cubes deep. That's a total of 1000 x 1000 x 1000 = 1,000,000,000 plastic cubes. The correct choice is B.