<ol>
<li><p>If f(x) is a parabola with a minimum at f(2)
which of the following is equal o f(6)?
A f(4)
B f(0)
C f(-2)
D f(-4)
E f(6)</p></li>
<li><p>At the Biltmore Hotel , 86 guests are in rooms with king size beds
and 57 guests are in rooms with bathtubs. If 83 of the guests are
in rooms with only a bathtub OR a king size bed, how many guests
are in rooms with both features?
A 60
B 56
C 36
D 30
E 27 </p></li>
<li><p>In the xy-coordinate plane, the point (2, r) is
a distance of 13 from the point (14, 2).
Which of the following could be r?</p></li>
</ol>
<p>the answer was -3 but i was wondering if there was another way to solve
this question other than using the distance formula and simplifying. </p>
<p>4.what is the volume of the smallest cube that could contain a sphere
with a radius of r?
A r^3
B 3r^3
C 4r^3
D 5r^3
E 8r^3</p>
<ol>
<li>The lowest score on the most recent chemistry exam in Professor
Wren's class was a 39, and the highest score was a 75.
Which inequality could be used to determine whether a particular
score, s, could have come from Professor Wren's class?
A |s-39|<= 75
B |s-20|<= 57
C |s-18|<= 55
D |s-60|<= 17
E |s-57|<= 18</li>
</ol>
<p>They’re all from PWNtheSAT’s diagnostic tests. @Syndekit Can you tell me how you did them? @PWNtheSAT you should make more tests, you’re good…and make one for writing.</p>
<p>If they’re right then I’ll explain.
For 6, make the diameter 2 (diameter is arbitrary) and find the area of the circle = pi.
With the diameter make two 90 45 45 triangles. The base of the shaded triangles is 2/root2 and the height of the triangles is the same. Find the area of one 90 45 45 triangle. This area is equal to both shaded triangles. The area is 1. Thus the answer is 1/pi or A.
For 7, the height is x and the length is y. The area of the entire figure is .5xy. Since BN is .2y, NC is .8y. Furthermore, the triangles are similar, so the height of triangle MNC is .8x. Using .5bh, area of the unshaded triangle is .32xy.
Finally, subtract the entire figure (.5xy) from the unshaded region (.32xy), and you get the shaded region which is .18xy or D.</p>
<p>^specific: It’s useful to think of absolute value of a difference as being the distance between points, in this case, the score s and another number.</p>
<p>If you visualize the top score of 75 and the lowest score of 39 on a number line, you know that s will definitely lie somewhere between those two (inclusive). Now take the midpoint of the range, which is 57. </p>
<p>What is the maximum possible distance of the score s from the midpoint? That has to be equal to the distance of 75 or 39 from the midpoint. Both are 18 away from the midpoint (75 is 18 higher and 39 is 18 lower). So the correct inequality is
| s - 57 | =< 18,
which is E, as posted above by treating in #4.</p>
<p>For 4, visualize a cube that just encloses s sphere of radius r. The sphere should be centered in the cube, and it should just touch the faces of the cube, at the centers of the faces of the cube. (In any other arrangement, the cube that encloses the sphere will have a larger volume.) The radius of the sphere is r. This means that the distance between any two opposite faces of the cube must be 2r. The distance is the same as the diameter of the enclosed sphere. So the side length of the cube is s = 2r. The volume of the cube is s^3 = (2r)^3 = 8 r^3, which is E, as people have posted above.</p>