Question

Model with Mathematics It can be very helpful to create simple models when you’re solving real-life mathematical problems. The model can often provide insight into how to understand the problem and can also help you decide what strategy to use. Consider the following example: Kelliann invents a new game that can be played outdoors. The playing field consists of a square with a half-circle connected to each side of the square so that the diameter of the circle is the length of the side of the square. If the side of the square is equal to nine feet, what is the total distance around the playing field for Kelliann’s game? Before starting to solve the problem, it’s a good idea to sketch a simple model so that it is clear what the playing field looks like. The playing field has three parts: a square and two half-circles, also called semicircles. The semicircles are joined along the left and right sides of the square. Your model might look like this: An illustration shows a square in the middle extending into two semi circles on the right and left sides. The 4 sides of the square are labeled 9 feet.It can be easier to understand the problem when you can see the relationship among the parts of the playing field. Use the model to compare your understanding of the problem with the problem’s description. To solve this problem, do the following: First, find the length around each semicircle. An illustration shows a square in the middle extending into two semi circles on the right and left sides. The 4 sides of the square are labeled 9 feet. The line thickness of the semi circles are thicker than the line thickness of the square.The formula for the circumference of a circle is C=2πr or C=πd , so the formula for the circumference of a semicircle is C=πr or C=(πd)2 . CC==(πd)2(9π)2 The length of each curved segment is (9π)2 feet. Then, add the circumferences together: (9π)2+(9π)2=9π feet Finally, look back at the model. There are two lengths you have not yet included in the distance around the outside of the figure. These are the two straight segments.An illustration shows a square in the middle extending into two semi circles on the right and left sides. The 4 sides of the square are labeled 9 feet. The top and bottom of the square, which connect with the semi circles, are drawn as dotted lines.The line thickness of the semi circles are thicker than the line thickness of the square.Add the lengths of the two straight segments; each is 9 feet long. So, the total distance around the outside of the playing field is 18+9π feet. Because pi is equal to about 3.14, the total distance around the playing field can also be expressed as about 46.26 feet. Cross Connections: Science Making models can be very helpful in science as well as in math. One example is DNA in genetics, which is often represented as a double helix or double spiral. Another is the structure of molecules in atomic theory. These are models of things we cannot see with the naked eye. Just as in math, scientific models can help people make sense of an idea. Just keep swimming! write it as a 6th grader and make it as if parents helped you without making it obvious

Answer 1

I drew a picture of the playing field: a square in the middle with a semicircle stuck to the left side and another semicircle stuck to the right side. Each side of the square is 9 feet, so each semicircle has diameter 9 feet.

Step 1 — curved parts:
Two semicircles together make one full circle. The circumference of a full circle is C = πd. Here d = 9, so the curved total = π × 9 = 9π feet.

Step 2 — straight parts:
The top and bottom edges of the square are still part of the outside. Each is 9 feet, so the straight total = 9 + 9 = 18 feet.

Add them:
Total distance around the field = curved + straight = 9π + 18 feet.

Approximate with π ≈ 3.14:
9π ≈ 9 × 3.14 = 28.26, so total ≈ 28.26 + 18 = 46.26 feet.

I drew the model and checked my steps at home so I’m confident the answer is 18 + 9π feet (about 46.26 ft).