| Problem |
Solution |
1) One angle of a triangle has a measurement of 70 degrees. The ratio of the other two
angles of the triangle is 5:6. What are the measurements of the other two angles?
|
1. Compute what the other 2 angles sum to:
180 - 70 = 110 degrees.
2. The smaller angle is 5/11 of this sum.
3. Find both angles =
(5/11) x 110 = 50 degrees and 110 - 50 = 60 degrees.
|
|
2) A group of scientists is duplicating cells for a science experiment. They observe that during each 10-minute period, one-quarter of the cells present died and the rest of the cells
all doubled. If there were 32 cells initially, how many cells were there after 40 minutes?
|
1. After 10 minutes: #cells: 32(3/4) x 2 = 48
2. After 20 minutes: 48(3/4) x 2 = 72
3. After 30 minutes: 72(3/4) x 2 = 108
4. After 40 minutes: 108(3/4) x 2 = 162 cells
|
| Problem |
Hint |
3) There are 832 steps from the basement to the restaurant level of the Space Needle. You start out going up 25 steps every 20 seconds. After one minute you slow down to
going up 24 steps every 20 seconds for the next minute, then 23 steps every 20 seconds
in the next minute and so on. If this pattern continues, how long to the nearest minute will
it take you to reach the restaurant?
|
- You start out at 75 steps per minute.
- After that you decrease by 3 steps every minute
- The progression is 75 + 72 + 69 + 66 + ..., which is:
- 75 +(75-3) + (75 - 6) + (75 - 9) + (75 - 12) + ...
- (75 + 75 + 75 ...) - 3(1 + 2 + 3 + 4 + ...n) = 832
- Let m = the number of minutes for your climb. Then:
- 75m -3(1 + 2 + 3 + 4 + ... + m-1) = 832
- The sum of consecutive numbers from 1 to n is = (1+n) n / 2
- In our case, n = m-1 so the sum is (m-1)(m)/2
- So the equation is 75m -3(m-1)(m) / 2 ≥ 832
The best way to find m is through guess-and-check
I'll give you your first guess:
Minutes
(m)
|
75 x m
|
-3(m)(m-1)/2
|
Total
steps
|
| 12
|
900
|
-3x12x11/2 =
-198
|
900-198=702
Too low!
|
| 16
|
1200
|
-360
|
1200-360=
840
|
The climb takes 16 minutes.
|
