The answer in itself is 1/128 and here is the procedure to prove it:
cos(A)*cos(60+A)*cos(60-A) = cos(A)*(cos²60 - sin²A)
<span>= cos(A)*{(1/4) - 1 + cos²A} = cos(A)*(cos²A - 3/4) </span>
<span>= (1/4){4cos^3(A) - 3cos(A)} = (1/4)*cos(3A) </span>
Now we group applying what we see above
<span>cos(12)*cos(48)*cos(72) = </span>
<span>=cos(12)*cos(60-12)*cos(60+12) = (1/4)cos(36) </span>
<span>Similarly, cos(24)*cos(36)*cos(84) = (1/4)cos(72) </span>
<span>Now the given expression is: </span>
<span>= (1/4)cos(36)*(1/4)*cos(72)*cos(60) = </span>
<span>= (1/16)*(1/2)*{(√5 + 1)/4}*{(√5 - 1)/4} [cos(60) = 1/2; </span>
<span>cos(36) = (√5 + 1)/4 and cos(72) = cos(90-18) = </span>
<span>= sin(18) = (√5 - 1)/4] </span>
<span>And we seimplify it and it goes: (1/512)*(5-1) = 1/128</span>
1) x=6, y=3
2) no solution
Let’s call the number of gallons of 30% solution “x”. If we mix together 40 gallons and x gallons of solution, we’re going to come out with a total of 40 + x gallons of that 20% solution. Mathematically, this is
40(12%) + x(30%) = (40 + x)(20%)
With the percents as decimals:
40(0.12) + x(0.3) = (40 + x)(0.2)
Simplifying, we have:
4.8 + 0.3x = 8 + 0.2x
Collecting the x terms on one side and the constants on the other:
0.1x = 3.2
And multiplying both sides by 10 to find x:
x = 32
So, you’ll need 32 gallons of 30% solution to yield 40 + 32 = 72 gallons of 20% solution.
Answer:
45
Step-by-step explanation:
45 +45=90
