Circular congruent holes with a diameter of 2 in. were punched out of a rectangular piece of paper with dimensions 8.5 in. by 11 in. What is the area of the paper that is left? Use 3.14 for [tex]$\pi$[/tex].
Answer (1)
Calculate the area of the rectangular paper: A rec t an g l e = 11 × 8.5 = 93.5 square inches.
Calculate the radius of each circular hole: r a d i u s = 2 2 = 1 inch.
Calculate the area of one circular hole: A c i rc l e = 3.14 × 1 2 = 3.14 square inches.
Calculate the total area of four circular holes and subtract it from the rectangle's area to find the remaining area: A re mainin g = 93.5 − ( 4 × 3.14 ) = 80.94 square inches.
80.94 in . 2
Explanation
Identify Given Information First, let's identify the given information:
The rectangular piece of paper has dimensions 8.5 inches by 11 inches.
There are four congruent circular holes punched out.
Each circular hole has a diameter of 2 inches.
We will use 3.14 as the value of π .
We need to find the area of the paper that remains after the holes are punched out.
Calculate Rectangle Area Next, we will calculate the area of the rectangular piece of paper.
The area of a rectangle is given by the formula:
A rec t an g l e = l e n g t h × w i d t h
Substituting the given values:
A rec t an g l e = 11 × 8.5 = 93.5 square inches.
Calculate Circle Radius Now, we will calculate the radius of each circular hole.
The radius is half of the diameter:
r a d i u s = 2 d iam e t er = 2 2 = 1 inch.
Calculate Area of One Circle Next, we calculate the area of one circular hole using the formula:
A c i rc l e = π × r a d i u s 2
Using π = 3.14 and r a d i u s = 1 :
A c i rc l e = 3.14 × 1 2 = 3.14 square inches.
Calculate Total Area of Circles Now, we calculate the total area of the four circular holes:
A t o t a l c i rc l es = 4 × A c i rc l e = 4 × 3.14 = 12.56 square inches.
Calculate Remaining Area Finally, we calculate the area of the paper that is left by subtracting the total area of the circles from the area of the rectangle:
A re mainin g = A rec t an g l e − A t o t a l c i rc l es = 93.5 − 12.56 = 80.94 square inches.
Final Answer Therefore, the area of the paper that is left is 80.94 square inches.
Examples
This type of problem is useful in manufacturing and design. For example, if you are designing a metal plate and need to cut out holes for bolts, you need to calculate the remaining area of the plate to determine the weight and cost of the material. This ensures efficient use of materials and cost savings.
