Help with my Math!!!!!!?

Question

Ok there are three problems I need to know how to do and the answer...... Even to help me with one would be fine......
1.
Between twelve o'clock midnight and twelve o'clock noon, how many times are the minute hand and hour hand at right angles to each other?
2.
Suppose it takes 3.25 miles along the highway to go from an altitude of 6000 feet. What is the percent grade?
*the grade of a highway is the vertical rise in the distance divided by the change in horizontal distance for the rise
3.
At some stops, a certain bus picks up 5 people. At other stops, it picks up 2, and at the same time, it lets go of 5. There are no other stops but these. It starts its run empty and picks up five people. At the end, it has 11 people aboard. If the number of stops is greater than 17, what is the least number of stops the bus makes?
Thank you!!!!

Answer 1

1. I would tackle this as 3:00 (which is a right angle) to 8:59:59 (just before a right angle) and then double the figure (to allow midnight to 2:59:59 and 9:00 to 11:59:59). There are 2 every hour (except for only one at 8-8:59). Therefore in the period between 3:00 and 8:59, there are 11. Double this, and your answer is 22. The reason it is not 24 as you would think is that the hour hand does not point directly to the number when the minute hand is pointing away from the "12". Therefore, although you would think that 3:30 should be a right-angle, it is not, because the hour hand is half-way between the 3 and 4. The right angle would occur at 3:32:43. Try it with an old clock!

Therefore, each right angle occurs every 32 minutes 43 seconds, or 32.7272 minutes, or every 0.54545454 hours. As you are trying to find the number of 0.54545454s in 12 hours, this is a straight divide - 12 / 0.545454 = 22.

2. You need to look at this like a triangle. The distance travelled is on the upward slope of the triangle, the altitude is along the vertical side of the triangle. You need to know the horizontal part of the triangle, which you are not given. Therefore to work this out, it is (h*h) + (v*v) = s*s - or in other words, horizontal squared plus vertical squared equals length of slope squared. Therefore by using algebra we could also write (s*s) - (v*v) = h*h.

This goes for any triangle with a right angle (and yours has that along the horizontal and then up the vertical). Try it with a triangle that has 4 along the horizontal, and 3 along the vertical - you get an answer of 5 along the slope. You can easily draw this to prove it, using 3,4 and 5 inches.

With your figures, s = 3.25 * 5280 (5280 is the number of feet in a mile) = 17160; v = 6000.
Therefore the sum is:
17160 * 17160 = 294465600.
6000 * 6000 = 36000000
294465600 - 36000000 = 258465600
The Square Root of 258465600 = 16076.865
h = 16076.865, therefore.

Now you have all the information you require, and it is a straight divide, the calculation is 6000 / 16076.865 = 0.3732 or 37.32%

3. Now this is one right up my street (see my screen name!). The calculation can be found by knowing the following :
Stops can either add 5 or subtract 3. Therefore, you need to find out how you can make up 11 by just adding 5 or subtracting 3 - but you need to do this in greater than 17 steps. You therefore need enough 5's added together to allow 3's to be subtracted.

Let's make this easy by creating a pattern :

3*5 = 15. We can not get to 11 by subtracting 3's
4*5 = 20. 3*3 = 9. This can be done in 7 steps, but too few
5*5 = 25. We can not get to 11 by subtracting 3's
6*5 = 30. We can not get to 11 by subtracting 3's
7*5 = 35. 8*3 = 24. This can be done in 15 steps. Not quite enough
8*5 = 40. We can not get to 11 by subtracting 3's
9*5 = 45. We can not get to 11 by subtracting 3's
10*5 = 50. 13*3 = 39. This can be done in 23 steps. Answer!

So you can see a pattern evolving. For every 3 stops where 5 people are picked up, you need to drop off 5 stops worth of people to get back to the original number. This is therefore 8 stops. The smallest way of getting to 11 is in 7 stops, and therefore the next way of getting to 11 is 15 stops, then 23, 31, 39 etc.


I hope that this helps in your homework

With Best Wishes

Adrian!

Answer 2

1. I want to say twice, 3 and 9 o'clock. But I wonder if the question wants you to include all the ones like 1:55, etc. Although in reality, that isn't exactly 90 degrees.
2. It's a steep 37.3%
a^2+b^2=c^2
c^2-a^2=b^2
17160^2-6000^2=258465600
sqrt of 258465600=16076
6000/16076=.373
3. 7 stops. 4 stops picking up 5 people gives you 20. 3 stops picking up 2 and dropping off 5 loses 9 people. 20-9=11.