## Monthly Math Challenge

From November thru February, TEAMS publishes a monthly math challenge designed to get students thinking and using the type of math they may encounter during the TEAMS competition.

Directions: Copy and distribute the math challenge to our students. They may work on the problem individualy or as a team, although each student's answer must be submitted individually by the TEAMS coach.

Correct entries for the month will be placed in a drawing and one name will be randomly drawn on the 2nd Friday of the month following the challenge. The student whose name is drawn will be sent a \$25 Visa gift card via their TEAMS coach.

Middle School Math Challenge – February

We want to power a single, incandescent bulb in a home using a solar panel on the roof.  Let’s assume our solar panel is 40% efficient (that is, 40% of the energy received by the solar panel is transferred into the home and can be used).

Our incandescent light bulb is 5% efficient: 95% of the power delivered to the bulb is lost as heat while 5% of the power received in converted to light.

1.  How much energy must be received at the solar panel to power our 100W light bulb for 30 minutes?
2.  How much energy is wasted by the panel and bulb?

High School Math Challenge - February

Background:

Voltage (V) Electrical potential (units:  Volts, or V)

Current (I) The flow of electricity (units:  Amps, or A)

Resistance (R) Characteristic of electrical components that resists current flow (units: ohms, or Ω)

Ohm’s Law V = IxR: the voltage across a component equals its resistance multiplied by the current through it.

Series Circuit elements in series require all current flowing through one element to also flow through the next element, and so on.

Parallel Circuit elements in parallel are connected to each other at both ends. They have the same voltage across them, but any current flowing into the parallel combination of these elements splits among them and joins back together after passing through.

Kirchhoff’s Voltage Law (KVL) The sum of all voltages around a path in a circuit must equal zero.

Kirchhoff’s Current Law (KCL) The sum of all currents entering a node must be zero; in other words, any current entering a node must also leave the node.

Node A junction of electronic components.

If you have added two batteries to a flashlight, you know the batteries need to be put in with the right polarity: each device should be labeled to show where the positive end should be pointed.  If the side of the battery says 1.5V and you add 2 batteries in the same direction, the total voltage is 3.0V – the voltage adds if the batteries (or power sources) are in series.

We have all seen batteries in flashlights, game controllers and many other devices.  The side of a AA battery says “1.5V”, meaning it can supply 1.5 Volts.  A 9V battery supplies – you guessed it:  9 Volts.

A solar panel array can be installed in series or in parallel.  If they are installed in series, the voltages add and the current through each must be the same.  For example, if we have 5 panels each capable of generating 9V (voltage) and 5A (current) in series, the total voltage will be 45V and 5A.  Note that each panel must be rated at 5A: adding one in series that is rated at a different current could damage the entire system.  If we put these same panels in parallel, we would have 9V and 25A.

Assume we have two different panels:

Panel A:  12V, 8A

Panel B:   9V, 4A

Find the total voltage and current available at the house for the following combinations of panels, or specify if the picture as shown would violate any of our above rules:

Combination 1:

Combination 2:

Combination 3:

Combination 4: For this combination, ignore our rules about current and find the voltage availabe given the configuration shown:

Make sure to answer all four questions in the submission.

Middle School Math Challenge – January

PlastyEarth is a startup with a goal to help the environment through engineering.  The company is developing new biodegradable plastics for packaging starting with an initial investment of \$10,000 of personal wealth from the founders. The new material was priced at \$5 per pound, as opposed to roughly 85 cents per pound of PET virgin bottle resin pellets. The company is developing a more efficient manufacturing process that will be fully scalable in approximately two years with an additional \$25,000, which is expected to lower the price to \$2 per pound. Anticipating the debt, the company calculated a projected cash flow of \$4000, \$6000, \$12,000, \$15,000, \$22,000, and \$40,000 for the next six years with a stable market interest rate of 5% with a cost of operations steady around \$5000 for two years, then \$2000 on the third and thereafter. Note that the material costs \$1.10 per pound to produce initially, but will drop to 88 cents after the third year.

1. What is the break-even point (in pounds of product) for the firm with the initial price?

Solution

High School Math Challenge – January

PlastyEarth is a startup with a goal to help the environment through engineering.  The company is developing new biodegradable plastics for packaging starting with an initial investment of \$10,000 of personal wealth from the founders. The new material was priced at \$5 per pound, as opposed to roughly 85 cents per pound of PET virgin bottle resin pellets. The company is developing a more efficient manufacturing process that will be fully scalable in approximately two years with an additional \$25,000, which is expected to lower the price to \$2 per pound. Anticipating the debt, the company calculated a projected cash flow of \$4000, \$6000, \$12,000, \$15,000, \$22,000, and \$40,000 for the next six years with a stable market interest rate of 5% with a cost of operations steady around \$5000 for two years, then \$2000 on the third and thereafter. Note that the material costs \$1.10 per pound to produce initially, but will drop to 88 cents after the third year.

1. What is the break-even point (in pounds of product) for the firm with the initial price?
2. What are the savings per pound of plastic if the new process is fully scaled?
3. What is the payback period to recover the debt of the initial investment and implementing the new manufacturing process if the payments on the process are \$5000 annually?

Solution

Middle School Math Challenge - December

Engineers have been tasked to implement a green roof on a building by planting greenery on a circular rooftop.  The entire surface cannot be used: there must be room left for maintenance.  The decision is made to use a hexagon shape for the greenery.

The circumference of a 3.00 m diameter circle is 9.42 m. The perimeter of a 4-sided figure (square) enclosed in the circle is 8.49 m.

Find both the perimeter and the area of a hexagon (6-sided) enclosed in the circle. (An answer for both perimeter and area must be provided.)

Solution

High  School Math Challenge - December

Engineers have been tasked to implement a green roof on a building by planting greenery on a circular rooftop.  The entire surface cannot be used: there must be room left for maintenance.  The decision is made to use a hexagon shape for the greenery.

The circumference of a 3.00 m diameter circle is 9.42 m. The perimeter of a 4-sided figure (square) enclosed in the circle is 8.49 m.

Solution

Middle School Monthly Math Challenge – November

A household pays 14 cents per kilowatt (kW) hour (hr) for electricity.  If the household leaves on a 30 Watts (W) compact fluorescent porch light for 3000 hours a year, what is the associated annual energy cost associated with this decision?

Solution

High School Monthly Math Challenge – November

A park is applying to be recognized as a dark sky park.  Currently, 71% of the park’s 2124 fixtures are dark sky compliant.  Over the next five years the park must have 90% of park lighting fixtures dark sky compliant.  How many fixtures must the park improve per year to meet their 90% dark sky compliance goal?

Solution