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DID YOU
KNOW?
2009-10 PROBLEMS OF THE WEEK:
Please turn problem solution in by Friday on a separate sheet of
paper with complete work and/or explanation. Correct solutions are worth an
extra credit homework!!!
Week 38 Last one of the Year!
How many squares would balance the third
scale? (Ignore distance from the middle of the scale.)
ANS. Add 2
squares. I think all of you that tried this got it! Congrats!
Week 37
Find positive
whole number values for A, B, and C such that (BC)2 = ABC. Note that
BC represents a two-digit number (where B cannot equal zero) and ABC a
three-digit number (where A cannot equal zero). Provide the value of A, B, and
C.
ANS. This
was too easy right? ABC was 625 with BC = 25!
Week 36
Two cyclists begin
a race against each other by starting in different towns. They will both travel
using the exact same route. Therefore, the cyclists will pass each other during
the race. If cyclist one is traveling at an average rate of 23 miles per hour
and cyclist two is traveling at an average rate of 25 miles per hour, how far
apart will the two cyclist be three minutes before they meet each other on the
route?
ANS. Only Ani turned
in a solution for this one! You just need to find the distance each
cyclist will travel in 3 minutes.
23/60 * 3 + 25/60 * 3 = 2.4
miles
Week 35
The power of the doubling effect....
Back in the 1970s, a commercial made
reference to a shampoo and the girl who tried the shampoo told
two friends, who in turn told to more friends the next day and
they also told two friends etc. The doubling effect is powerful
when it's gossip, lies and even handy to determine how many
times you'll need to rip a paper in half to come up with a
certain number of pieces.
Now, how many people would know
about the shampoo on the 10th day? The 20th day? How long before
a million people would know? Use exponential function to solve
this easy one!
ANS. After 10 days
1025 people had been told. After 20 days 1,048,576 had been told.
SO--using 1000000=2^x +1 yields x = 19.93 days! or after 20 days 1000000 people
know!
Week 34
A square whose
sides are each 10 cm and a right triangle with sides of 20, 21, and 29 cm
overlap so that the vertex of the right angle of the triangle is at the center
of the square. What is the area of the overlap?
Ans. The easiest way
to see this is to move triangle so that it hits two vertices of the square.
It must do this because of the right angle. Then you can easily see that
25% of the area is shaded and so area is 1/4 of 100 or 25. I believe only
Sumair got this POW!
Week 33
In the United
States, the U. S. Postal Service uses Zip Codes (postal codes) to route mail to
various regions on the country. Zip codes are five digit numbers that can begin
with zero. How many U. S. zip codes are possible such that the product of the
digits is even?
ANS? I think the
answer is 5^5. What do you think?!
Week 32
If the side of the larger square
above is 10 cm, find the sum of the perimeters of the 3 squares if the corners
of the next smaller square hits at the midpoints of the larger square.
Now imagine that the process of
nesting the squares continues forever. The sum of the perimeters
approaches a finite number. What is that number? (Hint: You
need to remind yourself about geometric series!)
Ans. Well this problem did
not drum up much interest! As I remember it Sana and Sumair only turned in
correct ans! You need to use the Geometric sum to infinity = a / (1 - r)
Sum of perimeter of 3
squares is 60 + sqrt20 and for 2nd part a = 40 and r = 1/sqrt2 so sum to
infinity = 80 + 40sqrt2!
Week 31
How many minutes
is it before 6:00 PM if 50 minutes ago it was four times as many minutes past
3:00 PM?
Ans. Aaron, Sana, Sumair, Zach H.,
and Bridget got it! If x = the number of minutes before 6PM then you can
use the equation
4x=180-x-50
and x = 26.
Week 30
For Calculus Students:
Integrate analytically from 0 to π, 1- cos2
x.
Ans. Only PJ turned
in a correct solution. You need to substitute using cos (2x) = 2cos2x
-1. Let's let PJ explain!
Week 29
For Precalculus Students:
Find the partial fraction decomposition:
(3x2
+ 4)
(x2
+ 1)2
Ans. 3 is numerator
for (x2
+ 1)2 and 1 is numerator for (x2
+ 1)! Several of you got this. Others got close!
Week 28
If [x] represents the
greatest integer function, then [x] represents the greatest integer less than or
equal to x. Find the value of x if the product of x and [x] equals 200.
Week 27
Abby, Sam, and Danni are three
contestants on a favorite game show. In one of the rounds of the game show the 3
contestants have a chance to spin a wheel up to two times in a row to try to get
the highest possible sum without going over 100. The wheel that they spin is
divided into 21 equal portions, each containing one element from the set of
numbers containing 1 and all of the multiples of 5 less than or equal to 100.
Part 1: Abby goes first. If she spins the wheel twice, getting a 15 and a 50,
what is the probability that on his first spin Sam gets a number larger than
Abby’s sum? Express your answer as a common fraction.
Part 2: Sam spins a 10 on his
first spin. If he spins the wheel a second time, what is the probability that
the sum of his two spins will be a multiple of 4? Express your answer as a
common fraction.
Part 3:After his second spin, Sam ended up with a
sum of 50. This means Abby was still in the lead. If Danni spins a number
greater than 65 on his first spin, he will not spin again. If he spins a number
less than or equal to 65 he will spin again. What is the probability that Danni
ends up being the winner? Express your answer as a common fraction.
ANS.
a. 15 + 50 = 65
Possible winning first spins: 70, 75, 80, 85, 90, 95, 100
7/21 = 1/3
b. Since Sam spun a 10 on his first spin,
the only possible sums he can get will end in a 5 or a 0 or will be 11. We know
that it is not possible for a multiple of 4 to end in 5. We also know that 11 is
not a multiple of 4, thus we are only concerned with numbers that end in 0 that
are also a multiple of 4. This tells us that the numbers we want are multiples
of 4(5) = 20. Sam could spin 10, 30, 50, 70 or 90 to end up with a sum that is a
multiple of 4. That is a probability of 5/21.
c. The probability that Danni wins on the
first spin is 1/3, as it was for Wendel in problem 1. Now we need to figure out
the probability of Danni winning in 2 spins. Let’s make a list. Notice that we
only look at first spins up to 65 because if Danni spun 70 or higher on his
first spin, he wouldn’t spin again.
That is a total of 7(13) + 8 = 99 two spin
combinations that would result in a win. That is a probability of 99/[(21)(21)]
= 11/49.
Thus, the probability that Danni beats Abby is 1/3 + 11/49 = 82/147!!!!
Week 26
You are in a
roomful of 36 people. Everyone is asked to shake hands with everyone. How many
handshakes will there be?
ANS. Well
this was a popular problem with many correct answers. How many of you did
36C2?
Or there was 36* 35/2? Some of you added up a list of numbers to get the
answer of 630.
Week 25
A. Last week
Kiplinger released its list of the 100 best values in public colleges.
Congratulations to The University of North Carolina at Chapel Hill (UNC) for
being at the top of the list! UNC charges about $4600 for in-state tuition and
fees. Kiplinger’s list may be more important than ever considering the average
tuition and fees at four-year public colleges have increased by 57% since five
years ago. According to this average increase, what would we estimate the cost
of in-state tuition and fees for UNC to have been five years ago? Express your
answer to the nearest dollar.
B. Not only
does the report indicate that UNC provides reasonable fees and great financial
aid packages, but it also boasts an accomplished student body. "Among students
in the freshman class of 2004-2005, 78% scored 600 or higher on the math
component of the SAT exam, and 73% scored 600 or above on the verbal section."
According to this information, what is the smallest possible percent of the
freshman class who could have scored 600 or above on the math and verbal
portions of the SAT?
C. Another
college receiving recognition in the article is the State University of New York
College at Geneseo. The article states this college has an enrollment of 5375
students and a student/faculty ratio of 19:1. According to this information, how
many faculty members does the college have? Express your answer to the nearest
whole number.
ANS. Nobody
got the B. part of this problem! A. was 4600/1.57 = nearest $2930. and C.
was x = 5375/19 = 283 people. The B answer is 51%. 22% is left
100-78 and then 73 - 22= 51%
Week 24
The final Algebra
exam in your school is a 50 question, multiple-choice, exam. You receive 1
point for each correct answer, 0 points for each omitted (skipped) question, and
a deduction of one-fourth of a point for each incorrect answer. You determine
that you need a final score of 36 points to get a final grade of an “A” in
Algebra. In how many ways on the exam can you achieve a final score that will
result in exactly 36 points? For each way, provide the number correct, number
incorrect, and number omitted on the exam to achieve 36 points.
ANS. A bunch of
you guys got this one!
| # Correct |
#Wrong |
# Skipped |
Grade
|
| 36 |
0 |
14 |
36 |
| 37 |
4 |
10 |
36 |
| 38 |
8 |
6 |
36 |
Week 23
Compute
the product of the following expression:
(1i1/100)(2i2/99)(3i3/98)
. . . (98i98/3)(99i99/2)(100i100/1)
Hint: i represents an imaginary number not a
variable i.
ANS. Many of you
quickly learned that the answer was just
i^(1+2+3+....+100).
Some of you forgot the the sum of an arithmetics series is
n(n+1)/2 or in this case 5050 and did it the hard way--added
up 100 numbers 1 to 100 on your calculator. Please
learn the easy way. Then because powers of i repeat in
fours you can figure that i^5050 = -1.
Week 22
Joe wants to have
a picnic with his girlfriend but at 2 pm the temperature is 97 degrees and too
hot. He knows that the temperature decreases exponentially between 2 pm
and 6 pm, and it cools to 90 degrees by 3 pm. If his girlfriend is only
comfortable when the temperature is at most 80 degrees what time will he be able
to start his picnic?
ANS. The answer
here was 4:34. No one even tried this one. You need exponential
equations and logs to find this.
Week 21
A square is inscribed in a circle and
is circumscribed about a smaller, concentric circle. What is the ratio of
the area of the inner circle to the annulus?
ANS. The ratio is one! This can be found
when you see the right triangle relationship between the 2 radii.
Week 19 & 20 Starting 2nd Semester!
This is a doosy! At what time between 4:00 and 5:00 do the hour and minute
hands point exactly in the same
direction?!
Ans. Note that the minute hand goes 6 degrees
in a minute and the hour hand .5 degrees. If the minute hand starts at 12
o'clock= 0 degrees and the hour hand starts at 4 o'clock = 120 degrees then
6x = 120 + .5x where x is the number of minutes
since 4 o'clock. Solve and x = 21 9/11 so time when hands are in exact
same position is at 4:21:49.09!
Many of you got this doosy!
Week 17 & 18
Given a + 1 = b + 2 = c + 3 = d + 4 = a + b
+ c + d + 5,
find the value of a + b + c + d.
Ans. By using substitution several times and
reducing to one variable c = - 4/3, b = - 1/3, a = 2/3, and d = -7/3 so there
sum is - 10/3.
Your work for this problem was messy!
Week 16
If y = 3x + 6, what is the minimum value of x3y ?
Ans. Substitute 3x + 6 in for y and get x^3(3x + 6).
This can be graphed to find the minimum or use the derivative = 0. The
minimum value is y= -5.0625 value when x = -1.5.
Week 15
If sin(x) + cos(x) = 1 and x is greater
than or equal to zero and less than or equal to two pi, determine all possible
values of x. Please give method of your solution.
Ans. Checking your unit circles you should
find that 0, π/2 and 2π work! How easy! There
were only about 4 solutions turned in for this easy one. Doesn't anyone
need the EC?
Week 14
Thanksgiving Dinner
Math - The Problems
In preparation for Thanksgiving dinner,
Mrs. G.Obble decided to get a new tablecloth for the dining room table. Her
rectangular table measures 3.5 feet by 6 feet, and she would like the tablecloth
to hang exactly 9 inches over each edge of the table. What is the area, in
square feet, of the ideal rectangular tablecloth for Mrs. G. Obble?
Now Mrs. G. Obble will turn her attention
to the food. According to an article she read, she should purchase 1¼ pounds of
turkey per person. She figures this is a good amount for an adult, but a child
would need only 2/3 of this amount. There will be 8 adults and 6 children at
Thanksgiving dinner. According to Mrs. G. Obble's logic, how many pounds of
turkey should she purchase?
When Thanksgiving has finally arrived, the
whole family sits down to a Thanksgiving dinner that includes turkey, mashed
potatoes, dressing, cranberry sauce and gravy. Uncle Bob’s favorite part of the
meal, though, is the olive tray. Mrs. G. Obble put the same number of green
olives and black olives on the tray, and Uncle Bob was the first person to
select from the tray. After Uncle Bob took eight green olives, the ratio of
green olives to black olives was 3:5. How many total olives were on the tray
before Uncle Bob took his olives?
ANS. I believe that 1st question was area of
37.5 square feet, 2nd question yielded need for 15 pound turkey, and 3rd
20 of each type of olive! You got these with arithmetic!
Week 13
Justin's family is having thanksgiving dinner at their house. there will
be a total of 9 people present for dinner. Justin and his cousin, Aaron,
are best friends and are insisting on sitting next to each other at the table.
If everyone sits at a big, round table in how many distinct orders can the 9
people sit? (Note: an order that can be created by rotating another order is not
considered different.)
Ans. With some
help from googling "circle permutation" many got this problem.
Counting Justin and Aaron as one, the circle permutation is 7! If Justin and
Aaron change places then their are twice as many arrangements or 2*7! = 10080.
Week 12
This problem involves blending 2 cups of
used paper (torn into little pieces) with a cup of water in a blender. (Yep, you
heard correctly - in a blender!) Then, dump the mixture into an 11" x 14" framed
window screen. After drying overnight, the mixture will form a "new" piece of
paper, which you can use to write a poem, draw a picture or make a paper
airplane. What size screen would you need if you had 8 cups of paper and 4 cups
of water blended? How many cups of paper and water would you need for a 21" x
22" screen?
ANS: Everyone that turned in a
solution this week got it correct! What were those answers?
Week 11
The student parking lot has 81 cars in it, all Acuras, Beetles, and Camrys.
There are half as many Acuras as Beetles and the number of Camrys is 80% of the
number of Acuras and Beetles together. How many of each kind of car is in
the parking lot?
ANS. Many of you know your percents! The answer was 15 Acuras, 30 Beetles,
and 36 Camrys!
Week 10
A man wants to paint the
floor of a Merry-Go-Round. It is formed by two concentric circles
(an annulus). He wants to determine the area of the floor (shown in
yellow in the figure below), so he will know how much paint to buy.
Because of all the machinery in the middle, he is unable to measure
the radii of the two circles. However, he finds the length of a
special chord to be 70 feet. This special chord is a chord of the
larger circle and a tangent to the smaller circle. (See diagram
below).
Can you determine the area of the
Merry-Go-Round which needs to be painted from just that one
measurement? Note: The area of an annulus is
π R2 - π r2. |
ANS. You need to
construct a right triangle that has sides r, .5 of tangent segment 70, with
hypotenuse of R. This gives r2 + 352 = R2
or R2 - r2 = 352. Since the area of the
annulus is
π R2 - π r2
multiply the equation by π and area will equal 352
π or 3848.
Week 9
Ans. This was a great week for POW! Many of you worked at the work
problem!
I believe the
answer was around 92 minutes. You should have arrived at this using the
equation: 1/2.5 + 1/4 = 1/ x or some similar rational equation.
Week 8
Find the value of x:
Ans. x = 20. Square both sides
and isolate just the x. You should see what to substitute to get the
answer.
Week 7
In early
2003, $2000 was deposited at a certain interest rate compounded annually. One
year later, $1200 was deposited in another account at the same rate. At the end
of that year, there was a total of $3573.80 in both accounts. What is the
annual interest rate?
(In
Spring 2008 only Cody got this. Jacob turned in an answer that was close but
wasn't correct because he used the continuous interest formula. Compounded
yearly interest formula is A = P(1+r)^n.)
Ans. The answer is 7%. Crista, Jacob,
and Kathy turned in correct answers. Sometime their solution was a little
inventive! You should have set up the following equation and solved for r!
$3573.80 = 2000(1+.01r)^2 + 1200(1+.01r)^1
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