Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2010 Grade 6 Geometry
Problem
Hint
1) A rolling bike wheel makes 2 full rotations and moves a distance of 13.6 feet. What is the diameter of the wheel to the nearest inch?
Use = 3.14 		1. Compute the wheel's circumference from the information about the rotations. C = _____
2. Compute the diameter D given this circumference. D = _____.
3. Round this to the nearest inch = _____
2) In the diagram AB has length 8 cm and M is the midpoint of AB. DC is 1/3 the length of AB. The area of isosceles triangle DMC is 8 sq. cm. What is the area of isosceles trapezoid ABCD?
From the area of the triangle DMC and its base DC,
compute it's height = _____ cm.
From this point, you can use one of 3 methods:

Method 1: Use the trapezoid area formula:
    A = (B1 + B2) H / 2
    A = _______

Method 2: Use area subtraction:
    Compute the area of the outer rectangle = _____ sq. cm.
    Each of the missing triangles is the same size as triangle DMC, so subtract 2 of those = _____ sq. cm.

Method 3: Add up the triangles:
    The area of triangles ADM and MCB are _____ sq. cm. each.
    Add these 2 areas to the area of DMC = _____ sq. cm.

Problem
Hint

3) Julius is making a duffel bag out of a piece of canvas that measures 3 feet by 4 feet. It will be cylindrical with dimensions 2 feet long and diameter 1 foot. Disregard the seams. The two end pieces are sketched out. Sketch in the third piece. How much canvas will be left over after he cuts out the pieces? Express your answer to the nearest tenth of a square foot. Use = 3.14		 The circumference of one of the ends is the width of the rectangle you need to make the duffel bag.
1. Compute the circumference of one of the end circles.
    C = _____
2. This is the width of a rectangle that makes the duffel bag. Draw this on the diagram.
3. Compute the area of the rectangular
    piece of canvas = ____ sq. ft.
4. Compute the area of one of the end circular pieces = _____
5. Take the area of the whole piece of canvas, subtract the rectangular piece and 2 of the end pieces to get the area of canvas left over = _____ sq. ft.

4) In the floor plan for a museum 8 revolving security cameras are shown as dots. They cover the whole floor plan. The owner wants to know: Can fewer cameras be used and where would they need to be mounted? Indicate number and location on the diagram on the answer sheet. 		The lines are the walls of the museum. Put dots on the diagram on the corners such that cameras placed there can "see" into all the corners. Make sure all corners can be seen by at least one camera and use as few dots as possible. Assume there is no limit on the range of the cameras.
5) Leo makes an octahedron by gluing together two pyramids along their bases. Its surface area is 16 square inches. Each triangle has a height of 2 inches. What is the surface area of one of the pyramids? 1. An octahedron has 8 triangular sides.
2. Divide the total surface area by 8 to get the area of one triangle = _____
3. A pyramid has a square base and 4 triangular sides.
4. Since you have the area of one of the triangular sides, all you need now is the area of the pyramid's square base. To get that you need to know the length of one of the trianglular bases. You can get that using the triange area and height information. Square base area = _____
5. Add this to 4 of the triangular areas to get the total pyramid area = _____