Problem 53 - Algebra and Trigonometry

Question

A modernistic painting consists of triangles, rectangles, and pentagons, all drawn so as to not overlap or share sides. Within each rectangle are drawn 2 red roses and each pentagon contains 5 carnations. How many triangles, rectangles, and pentagons appear in the painting if the painting contains a total of 40 geometric figures, 153 sides of geometric figures, and 72 flowers?

Step-by-Step Solution

Answer

The painting contains 30 triangles, 9 rectangles and 1 pentagon.

Step 1: Define the variables

Variables can be defined for each type of shape: Let \(T\) stand for the number of triangles, \(R\) for rectangles, and \(P\) for pentagons.

Step 2: Set up the equations

Set up three equations representing total figures, total sides, and total flowers. Equations will be as follows:\[T + R + P = 40\] (Each shape contributes one figure to the total)\[3T + 4R + 5P = 153\] (Triangles have 3 sides, rectangles 4, pentagons 5)\[0T + 2R + 5P = 72\] (A triangle has no flowers, rectangles have 2 roses, pentagons 5 carnations)

Step 3: Solve the equations

Solving for one variable in one equation, and substituting in others, will get the values of T, R and P. Subtracting third equation from the second gives:\[3T + 2R = 81\] Adding this to the first equation gives:\[4T + 2R = 121\] Dividing by 2, that gives the value of\[T = 30\] Substituting \(T = 30\) in the first equation from Step 2 will give \(R = 9\). And substituting values of \(T\) and \(R\) in that equation again will give \(P = 1\).

Step 4: Interpret the result

Based on the solved equations, the painting contains 30 triangles, 9 rectangles and 1 pentagon.