2) an engineer is trying to formulate the minimum cost set of materials to use in an upcoming project. there are four groups of materials, and exactly one item from each group must be selected. the costs are given below: group a cost group b cost group c cost group d cost a1 10 b1 2 c1 8 d1 5 a2 12 b2 1 c2 9 d2 8 a3 16 c3 11 d3 9 c4 13 if b2 is selected, then a3 must be used. we can use at most one of a1, b1, c1, and d1, while we must have at least two of a3, c3, c4, and d3. finally, d1 cannot be used unless a2 or c4 is used. formulate and solve a 0-1 integer programming model that will determine the optimal solution.

(n^3 - n)

= n(n-1)(n+1)

since it is divisible by 3 and 2 ,because it is a multiple 3 consecutive number

if, n-1 is odd then n+1 is also odd, then it is a multiple of 9

but if, n-1 is even then n+1 is also even, then it is a multiple of 4

so, largest multiple = max(9*2, 4*3) = max(18, 12) = 18

answer: 230 miles ( vote ! )

step-by-step explanation:

180.39 minus 17.99 equals 162.4

then divide 162.4 by .80 to get 230

the ratio of white tiles to blue tiles is 48: 16. the ratio in it's simplest form is 3: 1.

step-by-step explanation:

ratio: take the first number and add a colon next to it, then add the second number after that. make sure the ratio order matches up with the written problem!

simplest form: we can simplify the ratio 48 : 16 by dividing both terms by the greatest common factor (gcf).

the gcf of 48 and 16 is 16.

divide both terms by 16:

48 ÷ 16 = 3

16 ÷ 16 = 1

reducing to simplest form:

48 : 16 = 3 : 1

a) 7x=7

step-by-step explanation: