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\(A_{oct}=(\sqrt{\frac{(LE)^2}{2}}+EF+\sqrt{\frac{(FG)^2}{2}})\cdot (\sqrt{\frac{(FG)^2}{2}}+GH+\sqrt{\frac{(HI)^2}{2}})-\frac{4(LE)^2}{4}\\ A_{oct}=(1+1+1)\cdot (1+1+1)-2\\ \color{blue}A_{oct}=7\)

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Bruce bought four packages of cornbread mix that each weighs x pounds.     

He also bought a package of biscuit mix that weighs ? pounds more than one package of cornbread mix.     

If the total weight of the items that Bruce bought is 31 pounds, what is the weight, in pounds, of each package of cornbread mix?Enter an improper fraction instead of a mixed number     

Let's make a guess that ? would have been 1.     

     4x + (x + 1)  =  31     

     5x        + 1  =  31     

                  5x  =  30     

                    x  =  6     

_.     

All the sides of a triangle have integer length.   The perimeter of the triangle is 10, and the triangle is scalene.  How many such non-congruent triangles are there?     

Consider one side of the triangle, call it a.     

If a = 1, then b+c must equal 9.  2+7, 3+6, 4+5 all violate the Triangle Inequality Theorem.     

   a = 2, then b+c must equal 8.  1+7, 3+5 violate Triangle Inequality.     

   a = 3, then b+c must equal 7.  1+6, 2+5 violate the theorem.     

Try it for all lengths of a.  There are zero scalene triangles that satisfy the conditions.     

_.     

Where is the diagram below? Where is "down"?

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Ms. Lee has 12 boxes of crayons. Next year, she plans to buy 250% more boxes. How many boxes will she have in all?     

12 + (2.5)(12)  =     

12 + 30            =  42     

_.     

To win a game with a fair coin, you must flip n+1 heads in a row, where is the total number of tails flipped so far. So if your first flip is heads, you win and the game is over; if your first flip is tails but your next two in a row are heads, you win and the game is over; and so on. What is the probability that you win by flipping one head? Express your answer as a common fraction.     

It has to be heads on the first flip, so  1 / 2     

_.     

Jacob is constructing a fish tank that is 1.2 m long, 0.6 m wide, and 0.4m high. How much glass will he need to make it? (Note: There will be no glass on the top.)    

  1.2 x 0.6 x 1  =  0.72   bottom     

  1.2 x 0.4 x 2  =  0.96   front and back     

  0.6 x 0.4 x 2  =  0.48   left and right ends     

               total  =  2.16 m²     

_.     

In the figure PQ is parallel to RS. The length of QT is 4 cm; the length of TS is 6 cm; the length of PQ is 10 cm. What is the length of RS?​     

     TS : TQ  =  RS : PQ     

        6 : 4     =  RS : 10     

        6 / 4     =  RS / 10     

      60 / 4     =  RS     

             RS  =  15     

_.     

The answer is 1/4.