A band wants to distribute its music on compact discs (CDs). The equipment to produce the CDs costs $250, and blank CDs cost $5.90 for a package of 10. Which of the following represents the total cost, in dollars, to produce n CDs, where n is a multiple of 10? A. (250+0.59)n B. 250+0.59n C. (250+5.90)n D. 250+5.90n E. 250n+5.90
We know that there will be two components of the total cost: the equipment, and the blank CDs. The equipment is a fixed, one-time cost, because whether the band produces 10 CDs or 100 CDs, they will need the equipment to produce it, and the equipment doesn't increase the volume of CDs increases. So we know $250 is going to be part of the answer (no n attached to it), so we can rule out A, C, and E. The second element is the blank CDs, which cost $5.90 for a package of 10. It's easiest here to work through using an example. Let's say the band wants to produce 20 CDs - so n is 20 (and yes, it is a multiple of 10!). To calculate the total cost, we would use $250 + $5.90*2 (two sets of 10-CD packages). This is the same as $250 + $0.59*2, where n=2. So, we know that B is the correct answer!
Laird: Pure research provides us with new technologies that contribute to saving lives. Even more worthwhile than this, however, is its role in expanding our knowledge and providing new, unexplored ideas. Kim: Your priorities are mistaken. Saving lives is what counts most of all. Without pure research, medicine would not be as advanced as it is. Laird and Kim disagree on whether pure research: A. derives its significance in part from its providing new technologies B. expands the boundaries of our knowledge of medicine C. should have the saving of human lives as an important goal D. has its most valuable achievements in medical applications E. has any value apart from its role in providing new technologies to save lives
In this question, we need to identify which of the points (A-E) Laird and Kim disagree on with respect to pure research. Laird says that pure research is useful because it "provides us with new technologies that contribute to saving lives" and it "expand[s] our knowledge and provid[es] new, unexplored ideas" - where the latter is a more worthwhile result. Kim, on the other hand, says that "Your priorities are mistaken. Saving lives is what counts most of all", elaborating by saying medicine has advanced this far because of pure research. Since pure research saves lives through medical applications, D is the correct response.
Write a binomial to express the difference between the area of a rectangle with length p and width 2r, and the area of a circle with diameter 4r, where p>7h.
We know the area of a rectangle is length time width, so the area of the rectangle is p*2r = 2rp. The area of a circle is pi r^2 for a circle where r is a radius - the question gave us the diameter, and a radius is half of that. So, the area of the circle is pi*(2r)^2 = pi * 4r^2. To find the difference, it's helpful to find out which of the two areas is larger, so that we don't end up with a negative number. So they're telling us that p is greater than 7r. So since p>7r, 2rp>14r^2 (we've just multiplied both sides by 2r). We've done this so that the first term in the inequality is the same as the area of the rectangle. [Here I would draw out a diagram to show that the rectangle is bigger than the circle]. So the difference would be the area of the rectangle minus the area of the circle: 2rp - 4pi r^2.