Final Exam - Math 201

Transcription

1 Name: Final Exam - Math 201 Instructions: There are 14 problems on this exam, all with an equal weight of 20 points. Work any of the problems you like in any order you prefer. Indicate the 10 you wish graded by checking the boxes next to the appropriate problem numbers. Please show work unless otherwise indicated. If you have any questions...ask!! Good luck. 1. (20 points) The following problems concern the calculation of area. (a) (12 points) The following isosceles trapezoid has three congruent sides of length 5 2 and contains exactly two 45 angles. Compute its area. B C A D (b) (8 points) In the grid below, adjacent points are 2 inches apart. Calculate the area of the pentagon. 2. (20 points) Draw five distinct pentominoes that are also nets for an open cube. (There are 8 such pentominoes.)

2 3. (20 points) The following questions concern Euler s Formula. For each of them, either give an example for a positive answer or an explanation for a negative one. (If you give an example, you need not draw it...although you are welcome to if you feel more comfortable doing so.) (a) (5 points) Is it possible for a given polyhedron to have an odd number of faces, vertices, and edges? Why or why not? (b) (5 points) Is it possible for a given polyhedron to have the same number of vertices as edges? Why or why not? (c) (5 points) Is it possible for a given polyhedron to have the same number of vertices and faces? Why or why not? (d) (5 points) Is it possible for a given polyhedron to have the same number of faces and edges? Why or why not? 4. (20 points) Give the following formulas (you need not show work): (a) (2 points) The area of a triangle. (b) (2 points) The area of a square with side length s. (c) (2 points) The area of a circle with radius r. (d) (2 points) The area of a trapezoid. (e) (6 points) The area of a rhombus. (f) (6 points) The area of a regular hexagon with side length s. (Hint: A regular hexagon can be split into a certain number of very special triangles...)

3 5. (20 points) The following is a labeled copy of the tangram puzzle from your text. A D G C B E (a) (6 points) Give the best name for the pieces listed in the table below. F Piece Label Best Name D E F (b) (6 points) If you were asked to find all triangles in the figure, you would have to list A and B, but also A + B, the triangle formed by putting A and B together as in the figure. With this in mind, find all non-isosceles trapezoids in the figure. (There are 3.) (c) (8 points) If the area of the large square region is 2, give the fractional value for the pieces in the table below. Piece Label B D E F Fractional Area

4 6. (20 points) Below, the larger square has a side length twice that of the smaller square and the 4 pentagons shown are congruent. The area of the smaller square is 750 square feet. Give a brief explanation along with your answers. (a) (5 points) What is the area of the larger square? (b) (5 points) What is the area of one of the pentagons? (c) (5 points) What is the area of the right isosceles triangle? (d) (5 points) What is the area of the L-shaped region?

5 7. (20 points) (a) (8 points) In the following diagram, a b and c d. Find the measure of all of the labelled angles if m 1 = c d b a (b) (12 points) i. (4 points) Given an equilateral triangle with side length 4, compute its height h. (You need not show work.) ii. (8 points) Given a cube with side length h from above, compute the length s of the line connecting the back-left-top corner to the front-right-bottom corner of the cube. (You need not show work.) 4 4 h 4 h s h h h = s =

6 8. (20 points) a. (10 points) You and a friend have been captured by a particularly secret and mischievous government agency. Your friend has been placed in one of twenty cells. You have the ability to open any one of these cells, at which time all of the remaining nineteen will lock down and become impossible to open, so you must get it right the first time. You ask 7 of the guards for help, but each knows very little and exactly one lies. Their answers are as follows: Guard 1: Her cell is odd numbered. Guard 2: Her cell has a double digit number. Guard 3: Her cell number has at least one digit that is a 1. Guard 4: Her cell is even numbered. Guard 5: Her cell number is a multiple of 3. Guard 6: Her cell number is between 6 and 13. Guard 7: Her cell number is a factor of 12. In which cell, 1 through 20, is your friend? (Provide a brief explanation.) b. (10 points) If, in part (a), exactly one guard tells the truth, then in which cell, 1 through 20, is your friend? (Again, provide a brief explanation.) 9. (20 points) (a) (5 points) Name all of the regular polygonal shapes that can tessellate the plane by themselves. (b) (10 points) Explain why a regular pentagon cannot tessellate the plane by itself. (c) (5 points) Give an example of a pentagon that can tessellate the plane by itself. (A drawing would be nice, but a decent explanation will suffice if you do not trust your drawing skills.)

7 10. (20 points) (a) (10 points) One base of the trapezoid below is half the length of the other. How does the area of the smaller triangle compare to the area of the bigger one? (b) (5 points) As you know, two triangles are considered similar if they have the same three angles. Accordingly, two triangles are considered dissimilar if they do not have the same three angles. With this in mind, list all dissimilar isosceles triangles that contain a 40 angle. (c) (5 points) If the area of the smaller square below is 12, what is the area of the larger square?

8 11. (20 points) The following questions concern the identification of quadrilaterals by description. In each case, list the quadrilaterals that the description eliminates from the following list: square, rectangle, parallelogram, rhombus, kite, isosceles trapezoid, non-isosceles trapezoid. (a) (5 points) A pair of congruent sides and a pair of parallel sides. (b) (5 points) Two pairs of parallel sides. (c) (5 points) Two pairs of congruent sides. (d) (5 points) Exactly one right angle. (Be careful here...use some scratch paper.) 12. (20 points) Below is a triangle inscribed in a semicircle. How long is the altitude (dotted line) of the large triangle? 60 80

9 13. (20 points) Tennis balls are general sold three at a time and packed in a cylindrical container just big enough to store three tennis balls stacked in a line. Compute the volume of empty space in a fully packed can of tennis balls. (Hint: The container is three tennis balls high and one tennis ball across. Hence, if r is the radius of the tennis balls, the height and radius of the container are...) You may find the following formulas useful: Volume of a sphere: V s = 4 3 πr3, where r is the radius Volume of a cylinder: V c = πr 2 h, where r is the radius of the base and h is the height Your answer should be in terms of π and the radius r of the tennis balls. In other words, you should not use a calculator, and you should not replace π with Just leave it in terms of π and r. Don t simplify. 14. (20 points) Prove the Pythagorean Theorem (i.e., a 2 + b 2 = c 2 ) using any method you see fit. I provided two methods in class, but there are literally thousands of them, so I won t be picky. On the off chance that you prefer to use either of the proofs I showed you, I have provided diagrams to help you along.