A Simple Little Algorithm           Geometric Proofs Reconsidered

WHAT'S GOING ON?

SOME THOUGHTS ON WORD PROBLEMS

Trains, planes, ladders, orbits, discounts, screws, cones, money, boxes, triangles … Whether one likes math or not, dislike of “word problems” unites the two camps.

Why?

Marvin Gaye will tell us “What’s Going On?” as the presentation addresses, to me, the two main reasons word problems seem so hard (but don’t have to be).

 "If a chicken-and-a-half could lay and egg-and-a-half in a day-and-a-half, then how long would it take a couple of chickens to lay a dozen eggs?"   "As the earth rotates on its axis, how fast are we traveling (assume KC is 40° Latitude)?"

A FEW WORD PROBLEMS - AND A CHALLENGE

Send a few more ... It doesn't matter what grade-level or what level of toughness.

1. The length, in inches, of a box is 3 inches less than twice its width, in inches. If the perimeter of the box is 24 inches, what's the area of the box?

2. A boat departs Port Isabelle, Texas, traveling to an oil rig. The oil rig is located 9 miles east and 12 miles north of the boat’s departure point. About how many miles is the oil rig from the departure point?

3. In the standard (x, y) coordinate plane below, 3 of the vertices of a rectangle are (2,1), (-1,-1), and (6,-5). What is the 4th vertex of the rectangle?

4. The lead of a screw is the distance that the screw advances in a straight line when the screw is turned 1 complete turn. If a screw is 2 1/2 inches long and has a lead of 1/8 inch, how many complete turns would get it all the way into a piece of wood?

5. In the standard (x, y) coordinate plane, what is the distance of the line segment whose endpoints are (–3, 0) and (7, 4) ?

6. A 6-passenger plane cruises at 180 mph in calm air. If the plane files 7 miles with the wind in the same amount of time as it flies 5 miles against the wind, then what is the wind speed?

7. The upper Angel Falls, the highest waterfall on Earth, are 750 m higher than Niagara Falls. If each of the falls were 7m lower, the upper Angel Falls would be 16 times as high as Niagara Falls. How high is each waterfall?

8. What is the degree measure of the acute angle formed by the hands of a 12-hour clock that reads exactly 1 o’clock?

9. A bicycle wheel has diameter 1m. If the bicycle travels one kilometer, how many revolutions does the wheel make ?

10. Sales for a business were 3 million dollars more the second year than the first, and sales for the third year were double the sales for the second year. If sales for the third year were 38 million dollars, what were sales, in millions of dollars, for the first year?

11. A chord 24 inches long comes within 5 inches of the center of a circle. What is the radius of the circle?

12. An industrial cleaner is manufactured using only the 3 secret ingredients A, B, and C, which are mixed in the ratio of 2:3:5, respectively, by weight. How many pounds of secret ingredient B are in a 40-pound (net weight) bucket of this cleaner?

13. Samantha needs 20 quarts of 50% antifreeze solution in her radiator. She plans to obtain this by mixing some pure antifreeze with an appropriate amount of 40% antifreeze solution. How many quarts of each should she use?

14. Mike mows the front yard in 40 minutes, while Jack can mow the same yard in 50 minutes. How long would it take them if they worked together?

15. A neighborhood recreation program serves a total of 280 children who are either 11 years old or 12 years old. The sum of the children’s ages is 3,238 years. How many 11-year-old children does the recreation program serve?

DIFFUSION OF GASES

The following experiments were conducted to test her formula and to study factors affecting the rate at which gases diffuse.  When hydrogen chloride (HCl) and ammonia (NH3) vapors react, they form solid ammonium chloride (NH4Cl):

HCl(g) + NH3(g) → NH4Cl(s)

A swab soaked with HCl solution was inserted into one end of a glass tube (1 cm diameter), and, simultaneously, a swab soaked with NH3 solution was inserted into the other end, so that the swabs were 10 cm apart. The distance that each vapor traveled could be determined because, at the point they made contact, a white ring of NH4Cl formed. The reaction was done at different temperatures. The time it took for the ring to start to form and its distance from the HCl swab were measured for each trial.

Spent fuel (SF), a radioactive waste, is often buried underground in canisters for disposal. As it decays, SF generates high heat and raises the temperature of the surrounding rock, which may expand and crack, allowing radioactivity to escape into the environment. Scientists wanted to determine which of 4 rock types—rock salt, granite, basalt, or shale—would be least affected by the heat from SF. The thermal conductivity (how well heat is conducted through a material) and heating trends of the 4 rock types were studied.

Fifty holes, each 0.5 m across and 20 m deep, were dug into each of the following: a rock salt deposit, granite bedrock, basalt bedrock, and shale bedrock. A stainless steel canister containing 0.4 metric tons of SF was buried in each hole. The rock temperature was measured next to each canister after 1 year had passed. The higher the thermal conductivity, the more quickly heat is conducted through the rock and away from the canisters.

Rather Than Help --- These Hurt!