Problem 53 - College Algebra Essentials

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 consists of 19 triangles, 11 rectangles, and 10 pentagons.

Step 1: Identify and define the variables

Let \(x\) denote the number of triangles, \(y\) denote the number of rectangles, and \(z\) denote the number of pentagons. Each geometric figure contributes a specific number of sides and flowers to the painting.

Step 2: Formulate the equations based on the problem

There are a total of 40 geometric figures \(x + y + z = 40\). The total number of sides is 153 and each triangle, rectangle, and pentagon contribute 3, 4, and 5 sides, respectively, which gives us the equation \(3x + 4y + 5z = 153\). Each rectangle contains 2 red roses and each pentagon contains 5 carnations totaling 72 flowers \(2y + 5z = 72\).

Step 3: Solve the system of linear equations

First, isolate \(y\) in the third equation, \(y = 36 - 2.5z\). Substitute \(y\) into second equation \(3x + 4(36 – 2.5z) + 5z = 153\), which simplifies to \(x = 9 + z\). Substitute \(y\) and \(x\) into first equation \(9 + z + 36 - 2.5z + z = 40\), which simplifies to \(z = 10\). Now substitute \(z = 10\) into equations for \(x\) and \(y\) to find \(x = 19\) and \(y = 11\).

Step 4: Verify and interpret the solution

Verify the solution by plugging \(x = 19\), \(y = 11\), and \(z = 10\) into original equations to ensure they are all true. Therefore, the painting consists of 19 triangles, 11 rectangles, and 10 pentagons.