For each diagram, find the relationship between the number of shapes and the perimeter of the figure they form. Represent this relationship using a table, words, an equation, and a... For each diagram, find the relationship between the number of shapes and the perimeter of the figure they form. Represent this relationship using a table, words, an equation, and a graph.
Understand the Problem
The question requires analyzing geometric shapes and identifying patterns in the number of pentagons and hexagons in relation to their perimeter. It asks to represent this relationship using a table, an equation, and a graph.
Answer
The total perimeter for 2 pentagons and 1 hexagon is $16s$, and for 3 pentagons and 3 hexagons it's $33s$, represented by the equation $$ P_{total} = 6h + 5p $$.
Answer for screen readers
The total perimeter for 2 pentagons and 1 hexagon is $16s$. The total perimeter for 3 pentagons and 3 hexagons is $33s$. The general equation is $$ P_{total} = 6h + 5p $$.
Steps to Solve
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Identify the Shapes and Their Properties
We have pentagons and hexagons. A pentagon has 5 sides, and a hexagon has 6 sides.
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Outline the Number of Each Shape
From the diagrams:
- Problem 6: 1 hexagon and 2 pentagons
- Problem 7: 3 pentagons and 3 hexagons
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Calculate the Perimeter of Each Combination
For the calculations:
- For pentagons, the perimeter (P) is given by: $$ P_{pentagon} = 5 \times \text{side length} $$
- For hexagons, the perimeter is given by: $$ P_{hexagon} = 6 \times \text{side length} $$
Assuming the side length for both shapes is the same and denoting it as $s$.
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Determine Total Perimeter for Problem 6
For 1 hexagon and 2 pentagons: $$ P_{total} = P_{hexagon} + 2 \times P_{pentagon} $$ $$ P_{total} = 6s + 2(5s) = 6s + 10s = 16s $$
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Determine Total Perimeter for Problem 7
For 3 pentagons and 3 hexagons: $$ P_{total} = 3 \times P_{hexagon} + 3 \times P_{pentagon} $$ $$ P_{total} = 3(6s) + 3(5s) = 18s + 15s = 33s $$
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Create a Table to Represent the Data
Number of Pentagons Number of Hexagons Total Perimeter 2 1 $16s$ 3 3 $33s$ -
Formulate an Equation for Perimeter
Generalizing the perimeter based on the number of pentagons ($p$) and hexagons ($h$): $$ P_{total} = 6h + 5p $$
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Graph the Results
On a graph, plot the number of pentagons on the x-axis and the corresponding total perimeter on the y-axis based on the values calculated.
The total perimeter for 2 pentagons and 1 hexagon is $16s$. The total perimeter for 3 pentagons and 3 hexagons is $33s$. The general equation is $$ P_{total} = 6h + 5p $$.
More Information
The relationship between the number of pentagons and hexagons in relation to their perimeter shows how the perimeter increases with the addition of sides from different shapes. This can be visualized in a graph, demonstrating the linear relationship.
Tips
- Mixing up the formulas for calculating perimeter; ensure to use the correct sides for each shape.
- Forgetting to multiply the perimeter of a shape by its count when calculating the total perimeter.
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