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The figure above shows a solid formed by joining the bases of two square pyramids to opposite faces of a cube. If each edge of the solid has length inches, what is the total surface area, in square inches, of this solid
(A) 100 + 25√3
(B) 100 + 50√3
(C) 150 + 25√3
(E) 150 + 50√3
Choice (B) is correct. The surface of the solid is made up of 4 square faces of the cube plus the triangular (lateral) faces of the pyramids. Each face of the cube has an area of,(5)(5)=25 and so the 4 faces of the cube have a total surface area of 100 square inches. Each lateral face of the pyramids is an equilateral triangle, which can be divided into two 30°,60°,90° triangles by a perpendicular dropped from any of the triangle’s vertices. Each 30°,60°,90° triangle will have a base of 2.5, which is half of the side length 5, since the hypotenuse of the triangle is 5 and the triangle is 30°,60°,90°; the second leg of each triangle is 2.5√3. Since the area of a triangle is 1/2bh, the area of each triangle is
. Each lateral face of each of the pyramids is made up of two congruent triangles, thus the area of each lateral face of the pyramids is
. There are a total of 8 equilateral triangular faces on the solid, so the total area of triangular faces is
. Therefore, the total surface area of the solid is
The four points
lie in the xy -coordinate plane. If
are opposite sides of parallelogram
, what is one possible value of
The correct answer is any number greater than 3 and less than 5. Since
are opposite sides of a parallelogram, they must be parallel and equal in length. Since
are parallel, they are of equal slope, and so
, which simplifies to
are equal in length,
. This simplifies to
. The conditions
together are the same as the single condition
. Any fraction or decimal greater than 3 and less than 5 may be gridded as the correct answer.
, which of the following could be true?
(A) I only
(B) II only
(C) I and III only
(D) II and III only
(E) I, II, and III
Choice (E) is correct. To determine whether I, II, or III could be true, it is only necessary to find single examples of x and y that satisfy
while also satisfying the statement in question. An example of I being true is
. An example of II being true is
. An example of III being true is
If x and y are two different numbers selected from the integers from 500 to 1000, inclusive, what is the greatest possible value for
The correct answer is 1999. To achieve the greatest possible value for the expression, the numerator must be maximized, and the denominator should be the smallest positive number possible. Since x cannot equal y, and x and y are both integers, the denominator should be 1. Thus (taking x to be the larger number),
. In order to maximize the numerator, both x and y must be maximized. Since 1000 is the greatest value for either x or y, it follows that x must be 1000 and y must be 1000 - 1 = 999. Therefore, the greatest possible value of
Two functions f and g are defined by
, where a, b, and c are constants. If
, what is the value of a?
The correct answer is 3. Since
, it follows that
, it follows that
. Subtracting the corresponding sides of
, and so the value of a is 3.
The line graph above shows the number of students in the computer lab during seven periods on one school day. Each of the students in the computer lab during lunch was also in the lab during two of the other periods. What is the maximum number of different students who could have been in the computer lab on that day?
Choice (D) is correct. From the graph, the sum of the number of students who were in the computer lab periods throughout the day is
. Each of the 8 students in the computer lab during lunch was also in the computer lab during 2 other periods. These students were counted 3 times each (for a total of
). In determining the maximum number of different students who could have been in the classroom that day, each student can only be counted once. Therefore, students who were in the computer lab during lunch should be counted as 8 students instead of 24, and the sum 140 is at least
more than the number of different students. Since
, it follows that 124 is the maximum number of different students who could have been in the computer lab that day.
The figure above represents four offices that will be assigned randomly to four employees, one employee per office. If Karen and Tina are two of the four employees, what is the probability that each will be assigned an office indicated with an X ?
Choice (C) is correct because two of the four employees will have a chance to get the x box, thus the probability is 2/4 or 1/2 then that must be multiplied by the second probability which is now 1/3. The answer is 1/6