Assignment 4 (Due Thursday morning at the beginning of class)

1. Do the 4 problems below.

2. Study the Unit Reflections in S&S pp. 86-87 to make sure you know how to do such problems.

Problem 1. Equilateral Triangles

Suppose ABC is an equilateral triangle with each side s.

a) What is angle ABC in degrees? Explain. b) Let AF be an altitude of the triangle. What is angle BAF? Explain. c) What is the length |AF|? Explain. d) What is the area of the triangle?

Problem 2. More about equilateral triangles

It is a fact that all 3 altitudes of this equilateral triangle (still with side s) pass through a point P. giving a picture like the one below. [We aren't proving this now.]

a) Show that the triangles AFC and AGP are similar. What is the scaling factor? Use this scaling factor to figure out the length |GP|.

b) Here is another way to find |GP|. Triangles AGP and AGB have a common base AG. Draw the altitude segments of each triangle (using this base). Now there is a relationship between the ratio of areas of the triangles and the ratio of the altitudes. Use this to find |GP|.

Problem 3. Altitude and Volume of a regular tetrahedron

Build a regular tetrahedron from 4 equilateral triangles of side length s. Assume that triangle ABC is one face and that a point D is the fourth vertex. If you place ABC on a horizontal plane, D is directly above P (i.e., line PD is perpendicular to the plane of ABC so is perpendicular to all the lines AF, BG, CH).

a) Make a careful scale drawing of triangle APD (you can use graph paper). Use this to compute the altitude of the tetrahedron.

b) Use the facts you now know about the regular tetrahedron to compute its volume.

Problem 4. Ice Cream Cone Problem

Suppose you have an ice cream cone (a cone with a circular base). The height of the cone is H and the cone holds 100 cubic centimeters of ice cream. If you want to fill the cone partially with 50 cc of ice cream, how high (deep) will be the ice cream in the cone.

Comment. Being an ice cream cone, we put the vertex at the bottom and measure height up from there.

Hint. If any amount of ice cream is put in the cone (so that it is not mounded up, but the surface is a plane parallel to the base), then any blob of ice cream is a scaling of any other blob.