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What non-zero integer must be placed in the square so that the simplified product of these two binomials is a binomial: ?     

(3x + 12)(12x - ___)      

Place 48 in the space.    

The inside terms (12 times 12x) result in +144x.     

You need a number in the blank space to make 

the outside terms (___ times 3x) result in –144x.     144 / 3 = 48     

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Each gallon of paint covers 200 square feet. I have to paint one side of a wall that is 85 meters tall and 150 meters long. If a foot is approximately 0.3048 meters, then what is the smallest whole number of gallons I can buy and have enough paint to cover the whole wall?     

  85 meters / 0.3048 m/ft  =  278.87 ft     

150 meters / 0.3048 m/ft  =  492.13 ft     

so the surface of the wall is 278.87 ft • 492.13 ft  =  137,240.29 ft²     

the amount of paint required is 137,240.29 ft² / 200 ft²/gallon  =  686.20 gal     

you can't buy just 0.20 gallon, so you have to buy 687 gallons of paint     

_.     

A list of 10 positive integers, each less than 15, has an average of 6.     

What is the maximum number of ones (the number 1, not the digit) that could be in the list?      

Since the average of 10 numbers is 6, that means the total of those numbers is 60.     

If you have nine 1's, can you get the other 51 with one number.  No.     

If you have eight 1's, can you get the other 52 with two numbers.  No.     

If you have seven 1's, can you get the other 53 with three numbers.  No . . .  three 14's is only 42.     

If you have six 1's, can you get the other 54 with four numbers.  Yes . . .  three 14's and one 12..     

So the most 1's you can have is 6 of them.     

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Two rectangular boxes have the same volume.     

One box is a cube, and the other box had dimensions 8 ft by 4 ft by 12 ft.     

How many feet long is an edge of the cube?      

     Volume_rect  =  L • W • D     

     Volume_sqr   =  S • S • S     

     S • S • S  =  L • W • D     

              S³  =  8 • 4 • 12 ft³     

              S³  =  384 ft³     

              S    =  7.268 ft     

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If a, b, and c are positive integers satisfying ab + c = bc + a = 13 and ac + b = 41, what is the value of a + b + c?     

        a  =  8     

        b  =  1     

        c  =  5     ——–>>    8 + 1 + 5  =  14     

   check answer     

        (a • b) + c  =  13       (b • c) + a  =  13       (a • c) + b  =  41     

        (8 • 1) + 5  =  13       (1 • 5) + 8  =  13       (8 • 5) + 1  =  41     

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The square root of h is greater than 10/3 and less than 42/5.     

How many integer values of h satisfy this condition?      

       10/3   <  sqrt(h)  <  42/5     

        3.33  <  sqrt(h)  <  8.40     

       (3.33)²  <  h  <  (8.40)²     

        11.11   <  h  <  70.56     

       between 12 and 70 inclusive there are 59 integers     

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 In right triangle BCD with D = 90, we have BC = 9 and BD = 5. Find sin B.     

          sine = opposite / hypotenuse     

          sin(B)  =  DC / BC     

          Pythagoras c² = a² + b²     ——>>     b² = c² – a²     

                                                                   DC²  =  BC² – BD²     

                                                                   DC²  =  9²  –  5²  =  56     

                                                                   DC    =  sqrt(56)  =  7.4833     

          sin(B)  =  7.4833 / 9     

          sin(B)  =  0.8315     

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The number of diagonals in a certain regular polygon is equal to $2$ times the number of sides. How many sides does this polygon have?     

                                   d  =  s(s – 3) / 2     

                                   d  =  s(s – 3) / 2     

                                   2s  =  s(s – 3) / 2     

                                   2s  =  (s² – 3s) / 2     

                                   4s  =  (s² – 3s)     

                                   4s  =  s² – 3s      

                    s² – 3s – 4s  =  0     

                            s² – 7s  =  0     

                        s • (s – 7)  =  0     

                                      s  =  0    discard     

                                      s  =  7     

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The area of a shaded sector of the circle is 108pi square centimeters.     

Given an arc measure of 120, what is the length of the radius in centimeters?      

     Area of the 120^o sector = 108 π     

     120^o is 1/3 of a circle, so the area of the entire circle = 3 • 108 π     

          A  =  π r²     ——>>     π r²  =  324 π     

                                                 r²  =  324     

                                                 r    =  sqrt(324)     

                                                 r    =  18     

_.     

The number of diagonals in a certain regular polygon is equal to $2$ times the number of sides. How many sides does this polygon have?     

The number of diagonals in a polygon with s sides is d = s(s – 3) / 2.     

                                                                                    d  =  s(s – 3) / 2     

                                                                                   2s  =  s(s – 3) / 2     

                                                                                   2s  =  (s² – 3s) / 2     

                                                                                   4s  =  (s² – 3s)     

                                                                                   4s  =  s² – 3s      

                                                                                    s² – 3s – 4s  =  0     

                                                                                    s² – 7s  =  0     

                                                                                s • (s – 7)  =  0     

                                                                                              s  =  0    discard     

                                                                                              s  =  7     

_.     

Hey CInsbstnkng!

The number of diagonals of a polygon can be defined as \(d = \frac{n(n-3)}{2}\) where n is the number of sides (for any amount of sides over 3)

so then you can form an equation like so:

\(2n = \frac{n(n-3)}{2}\)

this sets 2 times the sides (n) against the number of diagonals.

Hope this helps!



(If you are interested in how to find the diagonal formula you can look here https://byjus.com/diagonal-of-polygon-formula)

The answer is 2*sqrt(7).

The answer is k = 15/7.

The answer is 12.

The solution is (-2, 5, 1, 17, -10).

The answer is 3.