<p>What is the radius of the largest sphere that will fit inside a cube with edges of length 2?</p>
<p>A)1
B) square root of 2
C) 3 square root of 2/2
D)2
E) 2 square root of 2</p>
<p>The integer x is the sum of three different positive integers, each less than 10. The integer y is the sum of three different positive integers, each less than 20. What is the smallest possible value of x/y?</p>
<p>In a certain high school, there are three times as many band members as orchestra members and twice as many orchestra members as jazz choir members. If no student can be enrolled in more than one of the three activities and there are 108 members in the three activities, how many students are in the band?</p>
<p>Could you give me the explanation on how you arrived to the answer?</p>
<p>1) The largest sphere that can be contained inside a cube, is one that has a diameter equal to the length of an edge of the cube. Here, the edge of the cube is 2, therefore the diameter is also 2, therefore the radius is 1.</p>
<p>2) If you want the smallest possible value of x/y, then you need Y to be the greatest number possible, and X the smallest number possible. Therefore, we want the sum of the least three different positive integers, each less than 10. Obviously, it is 1+2+3 = 6 Therefore, X = 6.
Now for Y, we want the greatest sum of three different positive integers less than 20. Therefore, Y = 17+18+19 = 54.</p>
<p>And now we can conclude that the smallest possible value of x/y is 6/54 which can be simplified to 1/9.</p>
<p>3) Let’s denote each group with a letter. Band members will be denoted as 3X, and Orchestra members will be denoted as X, and Jazz Choir members will be denoted as X/2.</p>
<p>Why do we denote them in this way? Orchestra members is X. Band members are three times Orchestra members, so they are 3 times X or 3X. And there are twice as many Orchestra members as Jazz Choir members, so Jazz Choir members are half of X or X/2.</p>
<p>Question stated that the sum of all these activities’ members is equal to 108.</p>
<p>3X + X + X/2 = 108. Therefore X = 24. Therefore the number of students in the band is 3 x 24 = 72.</p>
<p>Need any further elaboration?</p>
<p>I really appreciate it! Wow, very in-depth.</p>