(1) What is the number of units in the circumference of the circle with center at (−2, 3) and passing through (10, −2)? Express your answer in terms of π.
(2) By how many degrees does the measure of an interior angle of a regular octagon exceed the measure of an interior angle of a regular hexagon?
(3) Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction.
(4) What integer is closest to the area of a triangle whose sides are 5, 6 and 7 units?
(5) A triangle with sides 3a − 1, a2+ 1 and a2+ 2 has a perimeter of 16 units.
What is the number of square units in the area of the triangle?
(6) A figure skater is facing north when she begins to spin to her right. She spins 2250 degrees. Which direction (north, south, east or west) is she facing when she finishes her spin?
(7) What is the ratio of the area of a square inscribed in a semicircle with radius r to the area of a square inscribed in a circle with radius r ? Express your answer as a common fraction.
(8) The length of one side of a square is increased by 2 units and its other side is decreased by 2 units. By how many square units do the areas of the original square and the new rectangle differ?
(9) How many inches are in the area of a circle inscribed in a regular hexagon with side length 12 inches? Express your answer in terms of π.
(10) Twenty-eight circular pepperoni slices, each 1 inch in diameter, are placed on a circular pizza. The slices neither overlap nor hang off the edge. The diameter of the pizza is 14 inches. How many square inches of pizza are not covered by pepperoni slices? Express your answer in terms of π.
(11) If all angles are measured in degrees, the ratio of three times the measure of ∠A to four times the measure of the complement of ∠A to half the measure of the
supplement of ∠A is 3 : 14 : 4. What is the number of degrees in the measure of the complement of ∠A?
(12) The ratio of the length to the width of a rectangle is 12 to 5 and the area of the rectangle is 540 square units. What is the number of units in the length of the
rectangle?
(13) A 3” x 5” piece of paper can be rolled to form a cylinder by taping either pair of parallel edges together. What is the ratio of the volumes of the larger cylinder to the smaller cylinder obtained in this way? Express your answer as a common fraction.
(14) The surface area of a particular cube is 384 square centimeters. In cubic centimeters, what is the volume of the cube?
(15) How many triangles of any size are in the figure shown?
(16) Exactly forty-eight non-overlapping square tiles, each 1 inch by 1 inch, fit within a rectangle. What is the least possible number of inches in the perimeter of the rectangle?
(17) What is the ratio of the number of degrees in the complement of a
60-degree angle to the number of degrees in the supplement of a 60-degree angle? Express your answer as a common fraction.
(18) Each edge of a regular octahedron is colored orange or black. If every face of the octahedron has at least one orange edge, what is the smallest possible number of orange edges?
(19) A square is inscribed in a semicircle, and a second square is inscribed in the whole circle, as shown below. What is the ratio of the area of the smaller square to the area of the larger square? Express your answer as a common fraction.
(20) The measures of the three angles of a triangle are in a ratio of 4 : 5 : 6. What is the measure in degrees of the greatest supplement of these three angles?
(21) The surface area of a cube is 294 square centimeters. What is the ratio of the number of square centimeters in the surface area to the number of cubic centimeters in the volume of the cube?
(22) The graph shows six labeled points. How many distinct circles of radius 2 units are in the coordinate plane and pass through exactly two of the labeled points on this graph? (−1,1) (−1,−1) (1,1) (1,−1) (3,1) (3,−1)
(23) In △ABC, D is a point on AB such that CD = 6 and DB = 8. If ∠CAD = ∠BCD, how many units are in the perimeter of △ACD?
A B
C
D
(24) In circle O, AP = 2 cm, P O = 3 cm, and m∠BP O = 90◦. What is the
number of centimeters in the length of BP ?
O A P
B
(25) A 1 cm × 1 cm square is cut from each of the four corners of a square piece of cardboard with area 64 square centimeters. The sides are then folded up to make a rectangular box. What is the volume in cubic centimeters of the box?
(26) The rectangular region bounded by the lines with equations x = 1.2,
x = 2.6, y = −0.2 and y = d has area 14 square units. What is the greatest possible value of d? Express your answer as a decimal to the nearest tenth.
(27) The length of the diagonal of a square is 2√6 cm. What is the number of square centimeters in the area of the square?
(28) The coordinates of one of the endpoints of a diagonal of a rectangle are (−4, 2), and the coordinates of the point of intersection of the diagonals are (1, −1). The sides of the rectangle are parallel to the axes. What is the number of square units in the area of the rectangle?
(29) The vertices of a triangle are at (0, 0), (12, 0) and (18, 6). The triangle and its interior are rotated about the x -axis to form a solid figure. What is the volume, in cubic units, of this solid? Express your answer in terms of π.
(30) Pamela wants to make a quilt using fabric squares that are pre-cut to three inches on a side. One-fourth of an inch on each side is the margin for the seam and will be sewn under and out of view. How many of these fabric squares will she need to make a square quilt with side length five feet?
Answer Sheet
Number Answer Problem ID
1 26π 1BC3 2 15 5DD3 3 16/3 (inches) 22A5 4 15 44B5 5 12 square units 10B4 6 east 2322 7 2/5 01B4 8 4 33C3 9 108π 34C3 10 42π 5C001 11 70 degrees 4102 12 36 D355 13 53 4C55 14 512 cm3 A24C 15 56 triangles 3AC3 16 28 inches 33C2 17 1/4 0A4C 18 4 edges CA14 19 2/5 43A5 20 132 2DB3 21 6/7 B3AC 22 22 circles A102 23 18 55C3 24 4 0004 25 36 cubic centimeters BDC1 26 9.8 A5D3 27 12 4A001 28 60 3AB31 29 144π 35B5 30 576 CAA5
Solutions
(1) 26π ID: [1BC3]
No solution is available at this time.
(2) 15 ID: [5DD3]
The sum of the angle measures in a polygon with n sides is 180(n − 2) degrees. So, the sum of the octagon’s angles is 180(8 − 2) = 1080 degrees. The polygon is regular, so all the angles have the same measure, which means each is 1080◦
8 = 135◦. Similarly, the sum of
the angles of a hexagon is 180(6 − 2) = 720 degrees, which means each angle in a regular hexagon has measure 720◦
6 = 120◦.
Therefore, the desired difference is 135◦− 120◦ = 15◦ .
(3) 16/3 (inches) ID: [22A5]
Let the increase measure x inches. The cylinder with increased radius now has volume π(8 + x )2(3)
and the cylinder with increased height now has volume π(82)(3 + x ). Setting these two quantities equal and solving yields
3(64 + 16x + x2
) = 64(3 + x ) ⇒ 3x2
− 16x = x(3x − 16) = 0
so x = 0 or x = 16/3. The latter is the valid solution, so the increase measures 16/3 inches.
(4) 15 ID: [44B5]
(5) 12 square units ID: [10B4]
Sum 3a − 1, a2+ 1, and a2+ 2 to find 2a2+ 3a + 2 = 16. Subtract 16 from both sides and
factor the left-hand side to find (2a + 7)(a − 2) = 0 =⇒ a = −7/2 or a = 2. Discarding the negative solution, we substitute a = 2 into 3a − 1, a2+ 1, and a2+ 2 to find that the
side lengths of the triangle are 5, 5, and 6 units. Draw a perpendicular from the 6-unit side to the opposite vertex to divide the triangle into two congruent right triangles (see figure). The height of the triangle is √52− 32 = 4 units, so the area of the triangle is
1 2(6)(4) = 12 square units . 5 3 6 (6) east ID: [2322]
Each full circle is 360 degrees. Dividing 360 into 2250 gives a quotient of 6 with a
remainder of 90. So, she spins 90 degrees to her right past north, which leaves her facing east .
(7) 2/5 ID: [01B4]
Let s1 be the side length of the square inscribed in the semicircle of radius r . Applying the
Pythagorean theorem to the right triangle shown in the diagram, we have (s1/2)2+ s12 = r2, which implies s12=
4 5r
2. Let s
2 be the side length of the square inscribed
in the circle of radius r . Applying the Pythagorean theorem to the right triangle shown in the diagram, we have (s2/2)2+ (s2/2)2 = r2, which implies s22= 2r2. Therefore, the ratio
of the areas of the two squares is s
2 1 s2 2 = 4 5r 2 2r2 = 2 5 . r r (8) 4 ID: [33C3]
No solution is available at this time.
(9) 108π ID: [34C3]
No solution is available at this time.
(10) 42π ID: [5C001]
No solution is available at this time.
(11) 70 degrees ID: [4102]
Let x be the number of degrees in the measure of ∠A. Then we have 3x
4(90 − x) = 3 14,
from the information “the ratio of three times the measure of ∠A to four times the measure of the complement of ∠A is 3 : 14. Multiplying both sides by 23 and clearing denominators, we find 7x = 180 − 2x =⇒ 9x = 180 =⇒ x = 20. The measure of the complement of 20 degrees is 70 degrees.
Note: The hypothesis “if all angles are measured in degrees” is not necessary. The angle is determined uniquely by the given information regardless of the units used.
(12) 36 ID: [D355]
No solution is available at this time.
(13) 53 ID: [4C55]
No solution is available at this time.
(14) 512 cm3 ID: [A24C]
The surface area of a cube with edge length e is 6e2. Solving 6e2 = 384 gives
e =p384/6 = 8 centimeters. The volume of a cube with edge length 8 cm is (8 cm)3 = 512 cubic centimeters.
(15) 56 triangles ID: [3AC3]
No solution is available at this time.
(16) 28 inches ID: [33C2]
No solution is available at this time.
(17) 1/4 ID: [0A4C]
No solution is available at this time.
(18) 4 edges ID: [CA14]
No solution is available at this time.
(19) 2/5 ID: [43A5]
No solution is available at this time.
(20) 132 ID: [2DB3]
No solution is available at this time.
(21) 6/7 ID: [B3AC]
(22) 22 circles ID: [A102]
If two points are less than four units away, then there are exactly two circles of radius 2 containing both of them. If they are more than four units away, then no circles of radius 2 contain both of them. If the points are exactly four units apart, then there is one circle of radius 2 that contains both of them.
We would also like to know when a circle might contain three of these points. Because of this array, any three of these points (which are non-collinear) must form a right angle. If three points on a circle form a right angle, then two of them (the endpoints of the right angle) must form a diameter. This would imply that those two points are four units apart. It follows that if two points are fewer than four units apart, then none of the circles that contain both of them contain any other points on this grid.
We now label the points for clarity:
(−1, 1) (−1, −1) (1, 1) (1, −1) (3, 1) (3, −1) A D B E C F
The only pairs of points which are not fewer than four units apart are the following: AF CD AC DF
The other 62 − 4 = 11 pairs must be fewer than four units apart, producing 22 circles. The two pairs that are exactly four units apart, AC and DF, lie on a circle that also contains another one of the six points (E and B, respectively), so neither of these pairs contribute any circles.
Therefore, the total number of circles is 22 .
(23) 18 ID: [55C3]
No solution is available at this time.
(24) 4 ID: [0004]
No solution is available at this time.
(25) 36 cubic centimeters ID: [BDC1] No solution is available at this time.
(26) 9.8 ID: [A5D3]
No solution is available at this time.
(27) 12 ID: [4A001]
No solution is available at this time.
(28) 60 ID: [3AB31]
No solution is available at this time.
(29) 144π ID: [35B5]
No solution is available at this time.
(30) 576 ID: [CAA5]
No solution is available at this time.
