A small manufacturing company uses a quadratic function to model its daily profit as a function of the number of items that are manufactured in a day. The model predicts that the maximum daily profit will occur when $20 items are manufactured in a day.Quantity AThe daily profit predicted by the model for manufacturing 500 items in a dayQuantity BThe daily profit predicted by the model for manufacturing 550 items in a day
Answer(s): A
We are told the company uses a quadratic function to model daily profit as a function of items manufactured.For a quadratic model, the graph is a parabola.The maximum profit occurs at 20 items, so the vertex is at x = 20.Since the maximum is at 20, the parabola opens downward.Symmetry of a parabolaFor a parabola with vertex at x = 20, profit values at points equidistant from 20 will be equal.Example: profit at 10 and at 30 is the same.Compare 500 and 550Distance of 500 from 20 = 480Distance of 550 from 20 = 530Since 550 is farther from 20 than 500, and the parabola opens downward, the profit at 550 must be less than the profit at 500.
In a certain sequence, each term after the first term is equal to the preceding term plus the integer m. The 10th term in the sequence is 43, and the 20th term in the sequence is 83.Quantity AThe first term in the sequenceQuantity B3
General form of nth term nth term = first term + (n - 1)mLet first term = a10th term: a + 9m = 4320th term: a + 19m = 83(a + 19m) - (a + 9m) = 83 - 4310m = 40m = 4From 10th term:a + 9(4) = 43a + 36 = 43a = 77 > 3
x ≠ 0Quantity AThe average (arithmetic mean) of the numbers x, -x, and |x|Quantity Bx
Answer(s): D
If x is positive, the average is one-third of x and therefore less than x.If x is negative, the average is positive while x is negative, so it is greater than x.Because each case can occur, no consistent relationship holds. The relationship cannot be determined from the information given.
The area of the shaded triangular region is the arca of triangular region PQR.Quantity AQuantity BPS
Answer(s): B
Let's call the full area A.Since PQS and QSR make up the full triangle PQR,Since PQ = QR and triangle Q is isosceles, and S lies on PR, the altitude QS splits the triangle into two congruent right triangles. So, PS = SR, andArea of triangle PQS = area of triangle QSRBut that contradicts the earlier information that triangle QSR is 1/
In the figure, are APB is a semicircle with center O and CDEF is a square. The area of the semicircular region is .Quantity AThe area of region ADEBFCQuantity B3
Answer(s): C
Since CDEF is a square, DF and CE are diagonals with the same length that intersect at a 90 angle and bisect the 90 internal angles of the square, creating 45 angles at each vertex.ADO= ADC + CDO = 45°+ 45° = 90°ACO = ACD + DCO = 45°+ 45° = 90°DOC = 90°, A = 90°,Therefore, ADOC is a square.The same for EBFO.AO and OB are radiuses and the same time diagonals of squares ADOC and EBFO, consequently.DC and EF are diagonals of squares ADOC and EBFO, consequently.We can define the area of ADEBFC as the sum of areas of square CDEF and two triangles DCA and EBF.The area of CDEF is DC2, or 2.Triangles DCA and EBF are same congruent isosceles right triangles, where DC and EF are bases, so their area can be found using the formula:As two diagonals of a square cross at their midpoints, dividing each other into two equal halves, and height of triangle DCA is:So, the area of DCA isThe same for EBF.Therefore, the area of ADEBFC is:
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