Mr. Hoelzer's Class

You must solve these in order.
1) Find the sum of the odd integers from 13 to 107 inclusive.
2) Determine the number of sides a polygon must have if the sum of its interior angles is (your answer from #1)
3) The longest side of a triangle is (your answer from #2) while the shortest side is 1/2 of that same number. The angle between the two sides is 60 degrees. Find the third side.
4) Solve for a and b:
a3 + b3 = (your answer from 3)2
[ a cubed plus b cubed = (answer) squared]

Submit your answer by study hall Friday. You may submit as a pair if you wish. Be sure to include written or mathematical support for your work.

Driving west along route 66, Kevin saw a sign that read: Amboy 128 miles, Needles 202 miles. Later, a second sign indicated that the distance to Needles was twice the distance to Amboy. Later still a third sign indicated the distance to Needles was three times the distance to Amboy. How many miles apart were the second and third signs?

Each week a problem of the week will be posted for everyone to attempt. For extra credit, solutions are to be submitted by Study Hall on Friday.  Solution will be scored according to a rubric which will be available soon.

Sometimes a special problem will be posted in your specific classes page.  Be sure to check out your class page often.

To end the summer math sequence, I will present a problem similar to those you will see throughout the school year.  It may seem simple on face value, but be careful to understand logic and connections in the underlying math.  Usually in junior high math or science, you see the formula d=rt, for distance equals rate times time.  Here is a more thought provoking problem for the concept.

A family traveled in a car from town A to town B at a speed of 40mph.  On the way home from town B to town A, they traveled at 60 mph.  What was their average speed, in miles per hour, for the entire trip?  (Note: speed is the same as rate, and average speed is total distance/total time)  Support your answer with mathematics.  Send responses and questions to jhoelzer @2paws.net

The family was watching Deal or No Deal on GSN in a hotel room.  As you may know, a contestant chooses 1 of 30 cases and hopes it contains the $1M dollar prize; therefore, their probability of choosing the case is 1/30.  However, after that, a contestant completes a round by choosing a certain number of cases to remove from the game for that round.  There are many more than 30 possible ways a contestant can do this.  The order in which they choose the cases in a round does not matter because at the end of the round the same dollar amounts will be removed (much like a card hand in which the order the cards are dealt to you does not change the hand you will play).  In math, this is called a combination.  For example, in cards, a 5-card hand is a combination of choosing 5 cards from 52 (Notation:  ₅₂C₅ which is 2,598,960 hands).  Another example:  consider the letters a,b,c,d;  the total combinations of choosing 2 letters from 4, ₄C₂, is 6; ab, ac, ad, bc, bd, and cd.  I encourage the use of the internet to learn more about how combinations are determined.

Back to the Game:  After a contestant chooses their case, it is placed by him or her.  Then, a contestant’s decision consists of selecting the cases to remove in each round.  He or she has the following number of cases to decide to remove from the game in each round:

6 cases out of ?, then 5 cases from the remaining cases, then 4 cases, 3 cases, 2 cases, 1 case, 1 case, 1 case, 1 case, 1 case, 1 case, 1 case, 1 case, and finally the last decision is to keep your case or switch it with the last remaining case.  (ex.  The total # of possible decisions for round 1 is the total combinations of choosing 6 cases out of ?)

How many total possible decisions, ways to remove the cases, exist for a contestant who takes the game all the way to the end?

Now that number is why it is so hard to win a lot of money.

On vacation, we visited Epcot Center in Orlando, FL.  One of the rides is Mission Space to Mars.  When done, you walk out and can participate in a video game challenge.  In this challenge, a team of 4 astronauts works with individuals to be the first team to reach Mars by completing “repairs” to the ship.  Halfway through this challenge, the computer simulation gives a report of the percent efficiency of a team working together to complete the repairs.  In one such game I was watching, team Orion had an efficiency of 85% at the halfway point.  At the end of the game, their efficiency rate increased to 96%.  I paused and thought that seemed very difficult to accomplish if the team attempted close to the same number of repairs in each half of the game.  I have two questions:

a)       If a team attempts the same number of repairs in each half of the game, is the increase from 85% to 96% possible? Explain. Mathematics to prove your answer?

b)       If it is not possible in the above scenario, create a function or write a rule to represent the minimum number of successful repairs at 100% efficiency needed in the second half to raise a team’s rating from 85% to 96% independent of how many total attempts the team had in the first half.  Then, state the minimum number of attempts required in the second half of the game,  if in the first half, team Orion had 250 repair attempts.

I’m posting this early due to the holiday.

A few years back, we built a small deck for our swimming pool.  It is 54” high.  My wife went to Menards and bought standard pre-made stringers for a staircase.  The stringers had 5 steps cut into it with measurements of 11 inches and 7.5 inches that could be arranged so either measurement was the riser or tread of the steps.  However, to reach the deck, the stairs had to be placed with the 11 inches as the riser.  This creates a slope of nearly 1.5; too steep for steps, but we lived with it for three years.  This year proper steps will be built.

Your problem:  The slope of the steps is recommended between .5 and .8, exclusive, with a tread of no more than 12 inches.  Also, the staircase will have handrails parallel to the stringers on each side 30 inches above the steps.  Design a staircase to reach the 54 inch deck that meets the above criteria.  Create a side-view drawing of your design.  Label the riser, tread, total rise and run of the staircase, total length of the stringer (the board that goes from the deck to the ground on which the steps will be placed) to the nearest ½ inch, total # of steps, slope of the steps.    Send your measurements with design, if possible, to jhoelzer@2paws.net.

BONUS:  What is the angle of elevation of the handrails?

I have been performing general maintenance this summer on my tractors by changing all the filters and oils.  I was replacing the air filters on the Allis-Chalmers 8500.  When I opened up the box (square-based prism), the cylindrical air filter had on each end a piece of Styrofoam in the shape of a regular polygon to hold the filter from moving, to support the bases of the box on each side and keep it from squashing in during transport, and to allow the same box to hold different size filters.  The 1& ½ inch thick Styrofoam had a circle recessed 1 inch deep into it for the base of the filter to slide into.   In this case, the filter is 18 inches tall with a diameter of 10 inches;  the box is 11 by 11 by 19. If you were the engineer, decide what regular polygon you would use for the Styrofoam ends with reasoning.  Keep in mind you want it to perform well the tasks listed above yet use a minimum of Styrofoam (after all Styrofoam is made from petroleum). When done, state the side length of your regular polygon; then find the volume of Styrofoam used to create the ends (don’t forget it is recessed out to fit over the filter).  Send your results with support to my school e-mail.

OR    for younger students:

Find the surface area and volume of the box and air filter.  Send your results to my school e-mail.

I donated blood last Tuesday, my 21st unit (pint).  Below are some statistics about blood donations.

On Tuesday of  last week, Stephen Strasburg pitched his first major league baseball game for the Washington Nationals.  His fastball can reach 100mph, while his change-up averages 91 mph.  Let us define a  hitter’s reaction time as the time it takes the ball to leave the pitcher’s hand until it reaches home plate (or is hit).  What is a hitter’s reaction time for each of Strasburg’s pitches and what is the percent increase in reaction time a hitter has for his average change-up versus his top fastball?  (Assume the ball travels 60ft 6 in. from the pitcher until home plate, or is hit)  Send answers to jhoelzer@2paws.net