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Chapter 8: Problem 53

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?

Short Answer

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

Step by step solution

01

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.
02

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)
03

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 \(P = 1\).
04

Interpret the result

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

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