numbers: 20 * 1.6 = 32units: miles * km/mile ... combine into single numerator/denominator:
... so you have miles on top and bottom, and they cancel out (like 1/1=1), leaving km ... the units of your answer.
answer: 32 km
If you find yourself unsure where to start with these word problems, the best thing is to get some tutoring. Stop by my office hours, or make an appointment.
I. Unit conversion: One thing we will find ourselves doing a lot is converting a quantity from one set of units to another. For example, changing the distance from BG to Toldeo from miles to kilometers, or the distance from the Earth to the sun in light-minutes to kilometers.
Example: The distance from Point A to Point B is 20 miles. How many km is it? (Hint: there are about 1.6 km in a mile).
distance = (20 miles) * (1.6 km/mile)numbers: 20 * 1.6 = 32units: miles * km/mile ... combine into single numerator/denominator:
... so you have miles on top and bottom, and they cancel out (like 1/1=1), leaving km ... the units of your answer.
answer: 32 km
Now you try:
1) The distance from BG to Bloomington, IN is about 300 miles. Your Canadian friend wants directions in km. What is the distance in km?
2) Your Canadian friend has invited you to Toronto. He says its 390 km from the Ambassador Bridge in Detroit. How far is this in miles?
3) Your friend from Europa (the second large moon from Jupiter) wants to visit. You tell it the distance is 628 million km. It would like to know the distance in light-seconds. (Hint: there are 300,000 km per light second).
ANSWERS: (1) 480 km, (2) 243.75, or about 244 km, (3) 2093 or about 2100 light seconds.
II. Scaling: We sometimes make scale models of astronomical systems (solar systems, galaxies, Local Group) to help us envision the distribution of objects on a more human scale. This an example of a ratio problem.
Example: As a kid, I made a scale model of the America (the first yacht to win the "America's Cup" from the British). It had a 1:60 scale, meaning 1 inch on the model equals 60 inches (5 feet) on the original boat. If the original America was 130 feet long (that's a lie), how long was my model?
Then do algebra to isolate the model length on one side of the equation: Multiply both sides of the equation by the original length (orig_len) to get
On the left side, orig_len is on the top and bottom, so they cancel, leaving just mod_len on the top. So,
mod_len = orig_len / 60 = 130 feet / 60 = 2.167 feet
Things of that size we usually measure in inches, so let's convert:
mod_len = 2.167 feet * 12 inches/foot = 26 inches.
Now you try:
1) Pete on "Two Guys and a Girl" built a 1:80 scale model of a building that was 520 feet high. How high was the model?
2) We built a model of the solar system at 1:ten-billion (10^10) scale. If the actual sun is 1.39 million km in diameter, how large is the model sun in km? In centimeters? (HINT: the appendix of our text gives details on the metric system).
3) At that 1:ten-billion scale, Jupiter is the size of a marble (1.43 cm). How big is it in real life?
ANSWERS: (1) 6.5 feet, (2) 1.39 x 10-4 km or 13.9 cm, (3) 1.43 x 105 km.
III. Basic Equations: Distance = speed * time is a good example of the simple equations we will use in class. There are three variables (distance, speed, and time). In general, you will know the value of two of these variables, and will be asked to find the value for the third.
Example: Your Canadian friend is driving to Bloomington (300 miles) at 60 miles/hour. How long will it take him?
In this case, we want to find the time. The first thing to do is perform algebra on the equation to isolate the variable you seek on one side of the equation. In this case, you would divide BOTH sides of the equation (thus maintaining the BALANCE implied by the equals sign) by speed:
and the speed/speed on the right side cancels to one (1). Thus,
(A = B is the same as B = A ... the BALANCE is the same). Then you just plug in the numbers for distance and speed:
numbers: 300 / 60 = 5
units: here is the trick: if you have a fraction on the bottom of a fraction, like miles/hour above, you can put it on top of the fraction by inverting it:
and the miles/miles cancel out, leaving hr on the top ... the units of our answer are in hours (good, a unit of time!)
answer: 5 hours.
You try:
1) You are driving to Toronto from Detroit, 390 km. The legal limit on the highway is 100 km/hr. How long will it take?
2) You and your Canadian friend drive to his cottage 200 km north of Toronto. He says it will take 5 hours. What will be your average speed (and what kind of roads do you think he is taking you on!)?
3) After that ride, you think your Canadian friend is a little nuts. Your map shows a fairly direct route back to Detroit. You figure if you can go 60 km/hr you will be back in your Home Country in 6 hours. How far is it in km? In miles?
ANSWERS: (1) 3.9 hours, (2) 40 km/hr or 25 mi/hr ... pretty rough roads, or lots of traffic!), (3) 360 km or about 223 miles.
Sometimes, you need to do some unit conversion within this kind of problem to make the units match up, so they can cancel.
Example: After profuse appologies, your Canadian friend coaxes you back to Toronto for a visit. Not liking all this metric nonsense, you drive around south of Lake Erie to Buffalo and cross at Niagra Falls. The sign at the border says "Tronoto 100 km" and your speedometer says 75 miles/hr (they fly on the Queen Elizabeth Way!). How soon will you be there?
But the units don't match ... the km won't cancel with the miles. You need to convert km to miles (or miles to km). Fortunately, you remember your annoying ASTRONOMY class where you learned 1 km = 0.62 miles (0.62 miles for each km, or 0.62 miles per km).
Now on the top of the equation, km/km cancel out, and top-to-bottom, mi/mi cancels out (its like 1/1), leaving 1/hr on the bottom. We learned earlier that 1/hr on the bottom is like hr/1 on the top, or just hr. That's our units... hours. The numbers work out to 0.83, so 0.83 hours. Multiply by 60 min/hr to get 50 minutes.
You try:
1) You have such a good time with your Canadian friend that you leave late for home. Your poor old car can just manage to go 90 mi/hr. How soon will you make it to the border, 100 km away?
2) Your friend from Europa says its spaceship can travel at 1/10 the speed of light. How long will it take it to travel to 32 light-minute distance to Earth? (Hint: speed of light is 300,000 km/sec)
3) Your friend from Europa joins you on a "road trip" to Toronto, but you take its spaceship. It travels the 320 miles in 4.2 seconds. Compare this speed to the speed (1/10 light-speed) your Europan friend bragged about in question 2. Any thoughts on why it went a different speed?
ANSWERS: (1) 0.69 hours or about 41 minutes, (2) 2.09 x 104 sec, or about 349 minutes, or about 5.8 hours, (3) 123 km/s, which is much less than the 30,000 km/s it bragged about for the Jupiter to Earth trip. Perhaps it has to go slower in earth's atmosphere, compared to the empty vacuum of space, to avoid overheating by friction with atoms in the gas.
See me if you have any questions or want more practice problems.
