How many circles with a diameter of 4 inches can be cut from a cartolina if the cartolina has a length of 35 inches and a width of 10 inches?
The correct answer and explanation is:
Let’s analyze the problem step-by-step.
Given:
- Diameter of each circle = 4 inches
- Cartolina (sheet) dimensions = 35 inches (length) × 10 inches (width)
Step 1: Understand the problem
You want to cut as many circles as possible from the rectangular cartolina. Each circle has a diameter of 4 inches, which means the radius is 2 inches.
The goal is to find the maximum number of circles of diameter 4 inches that can fit inside the rectangle of 35 × 10 inches.
Step 2: Determine how many circles fit along each dimension
Since the diameter of each circle is 4 inches, the circles must be placed without overlapping, so the center-to-center distance horizontally and vertically is at least 4 inches.
- Along the length (35 inches): Number of circles that fit = $\left\lfloor \frac{35}{4} \right\rfloor = \left\lfloor 8.75 \right\rfloor = 8$ circles
- Along the width (10 inches): Number of circles that fit = $\left\lfloor \frac{10}{4} \right\rfloor = \left\lfloor 2.5 \right\rfloor = 2$ circles
Step 3: Calculate total number of circles
Total circles = circles along length × circles along width = 8 × 2 = 16 circles
Final answer:
You can cut 16 circles with a diameter of 4 inches from the 35 inch by 10 inch cartolina.
Explanation (around 300 words):
The problem asks for the maximum number of 4-inch diameter circles that can be cut from a rectangular sheet of cartolina measuring 35 inches long and 10 inches wide.
Each circle has a diameter of 4 inches, which means that each circle occupies a square space of 4 inches by 4 inches when arranged in a grid pattern without overlapping. To find out how many such circles fit into the rectangle, we divide the length and width of the cartolina by the diameter of the circles.
First, dividing the length (35 inches) by the diameter (4 inches) tells us how many circles fit horizontally. Since 35 divided by 4 is 8.75, only 8 full circles fit along the length because partial circles cannot be counted.
Next, dividing the width (10 inches) by the diameter (4 inches) tells us how many circles fit vertically. Since 10 divided by 4 is 2.5, only 2 full circles fit along the width.
By multiplying the horizontal and vertical counts (8 × 2), we find that a total of 16 circles can be cut from the sheet.
This approach assumes the circles are arranged in a simple rectangular grid pattern. While a hexagonal (honeycomb) packing can sometimes fit more circles into a given area, that method is more complex and usually not considered unless specified. For practical cutting, arranging circles in rows and columns is standard and ensures minimal material waste with simple layout planning.
Therefore, 16 is the maximum number of circles of diameter 4 inches you can cut from a 35 inch by 10 inch cartolina sheet.