1. Diagram two real world binary problems in two different fields using applications of Boolean Algebra.

Boolean function (𝑥, 𝑦, 𝑧) is given by formula represented in the table below.

• Represent the formula for the function (𝑥, 𝑦, 𝑧) in graphical form (i.e. by circuit). Find the depth of the formula as the height of binary tree constructed for its calculation.
• Construct value table (truth table) for the function 𝑓(𝑥, 𝑦, 𝑧)
• Construct PDNF and PCNF for the function (𝑥, 𝑦, 𝑧) with a help of its value table. Find lengths of these PDNF and PCNF. 4) Construct ANF (Zhegalkin polynomial) for the function (𝑥, 𝑦, 𝑧) with a help of its PDNF or using undefined coefficient method. Determine essential and fictitious variables of the function (𝑥, 𝑦, 𝑧).

 .                  𝑓(𝑥, 𝑦, 𝑧) = ((𝑧 → (𝑥 → 𝑧)) ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ↔ 𝑦) ∨ ((𝑥 → ̅𝑧 ) & (𝑧 → 𝑥))

1.)  Some punch is made by mixing grape juice and bubbly water in a ratio of 4 to 3. Give at least five different pairs of quantities of grape juice and bubbly water that you could use to make the punch mixture in that ratio. Include at least two examples that involve fractions or mixed numberquantities. Show or explain why all the mixtures are in that same ratio of 5 to 2 using a method your choice (i.e. double-sided number line, strip diagram, batch table).

2.)  Using a ratio of 3 to 8;

a.)  Give two different ways of using this ratio with different ‘part-to-part’ perspectives.

b.)  Give two different ways of using this ratio with a ‘part-to-whole’ perspective.

3.)  A drink mixture is made by mixing cola and lemon-lime soda in a ratio of 5 to 3. How much cola and how much lemon-lime soda will you need to make 240 cups of the drink mixture?  Complete either (a) OR (b).

(a) Show how to solve the problem from the multiple-batches perspective by reasoning about multiplication and division with quantities. You can support your work with a table or double-sided number line.

(b) Show how to solve the problem from the variable-parts perspective by reasoning about multiplication and division with quantities. You can support your work with a strip diagram.

4.)  Traveling at a constant speed, a race car goes 15 miles every 6 minutes.  Use the unit rate method to solve the following:  (a) How far does the race car go in 15 minutes?  (b) How long does it take the race car to go 20 miles?

5.)  A type of fertilizer is made by mixing nitrogen and phosphate in an 8 to 3 ratio. For 35 kilograms of nitrogen, how many kilograms of phosphate are needed to make that type of fertilizer?

6.)  To make jewelry, jewelers often mix gold and copper in a 7 to 5 ratio. How much gold should a jeweler -mix with 42 grams of copper?​

7.)  You need 3 cups of grape juice for a recipe that makes 4 servings. How many cups of grape juice will you need if you want to make 10 servings of the same recipe?

8.)  Consider all ramps that could be built for which the ratio of height to length is 2 cm to 7 cm. Let x be the unspecified length of a ramp in feet, which can vary, and let y be the corresponding height of the ramp in feet. Derive and explain an equation of the form y = c•x for a suitable constant c (which you should find).