Alex wants to cut out as many circular badges as possible from a rectangular
sheet of cardboard(100cm by 60cm). The diameter of each badge is 14 centimeters. Use 22 an approximation for pi.
a) How many badges can Alex cut?
b) What is the area of the cardboard left over?
pi is actually about 22/7
This problem is a lot harder than it looks because you should really pack the discs in a hexagonal pattern but I suspect the best we can do is put squares around the circles.
100 /14 = 7.14 so 7 lengthwise
60/14 = 4.28 so 4 that direction
7 * 4 = 28
now
total area = 6000 cm^2
area of 28 circles = 28 (22/7)(49)
then subtract
To find the number of badges Alex can cut, we need to determine how many rows and columns of badges can fit on the rectangular sheet of cardboard, and then multiply those numbers together.
1) Calculate the number of rows that can fit on the rectangular sheet:
- It is given that the diameter of each badge is 14 centimeters, which means the radius is half of that, so the radius is 7 centimeters.
- The sheet of cardboard is 100 centimeters wide, so divide this by the diameter of each badge to determine how many badges can fit in one row: 100 cm / 14 cm = 7.14 (rounded down to 7)
- Therefore, the number of badges that can fit in one row is 7.
2) Calculate the number of columns that can fit on the rectangular sheet:
- The sheet of cardboard is 60 centimeters long, so divide this by the diameter of each badge to determine how many badges can fit in one column: 60 cm / 14 cm = 4.29 (rounded down to 4)
- Therefore, the number of badges that can fit in one column is 4.
3) Multiply the number of rows by the number of columns to find the total number of badges that can be cut: 7 rows x 4 columns = 28 badges.
Therefore, Alex can cut a total of 28 badges.
To find the area of the cardboard left over, we need to subtract the area of all the badges from the total area of the rectangular sheet.
1) Calculate the area of one badge using the formula for the area of a circle: A = πr^2, where r is the radius.
- Given the diameter of each badge is 14 cm, the radius is 7 cm.
- Use the approximation of π as 22.
- Calculate the area of one badge: A = 22 x 7^2 = 22 x 49 = 1078 cm^2 (approx.)
2) Calculate the total area of all the badges by multiplying the area of one badge by the total number of badges: 1078 cm^2 x 28 badges = 30,104 cm^2 (approx.)
3) Subtract the total area of the badges from the total area of the rectangular sheet to find the area of the cardboard left over: 100 cm x 60 cm - 30,104 cm^2 = 5,896 cm^2 (approx.)
Therefore, the area of the cardboard left over is approximately 5,896 cm^2.
Well, Alex certainly has a round of work ahead of them! Let's figure out the answers with a touch of mathematical comedy:
a) To determine the maximum number of badges that can be cut, we need to calculate how many can fit in each row and column. Taking the diameter of each badge, which is 14 centimeters, we can fit approximately 5 badges along the width (60 cm) and approximately 7 badges along the length (100 cm). Multiplying these numbers together, we get 5 * 7 = 35 badges in total. So Alex can cut out a whopping 35 circular badges!
b) Now let's find out how much cardboard is left over after the badge extravaganza. The area of the rectangular sheet is 100 cm * 60 cm, which gives us 6000 square centimeters. Since each badge has an area of (22/7) * (7^2) = 154 square centimeters (approximating pi with 22), the total area covered by the badges is 35 * 154 = 5390 square centimeters. To find the area of the cardboard left over, we simply subtract the area covered by the badges from the total area of the sheet: 6000 - 5390 = 610 square centimeters.
So, Alex can cut out 35 badges and will be left with a rectangular piece of cardboard measuring 610 square centimeters in area. They'll just have to find another crafty project for the leftovers!
To find the number of badges Alex can cut and the area of the cardboard left over, we need to calculate the maximum number of badges that can fit in the given rectangular sheet of cardboard.
First, we need to calculate the area of one badge. The formula to calculate the area of a circle is given by: A = πr^2, where A is the area and r is the radius.
Given that the diameter of each badge is 14 cm, the radius (r) is half the diameter, which is 7 cm.
a) To calculate the number of badges, we need to divide the area of the cardboard by the area of one badge.
Area of one badge = πr^2
Area of one badge = 22 * (7 cm)^2
Area of one badge = 22 * 49 cm^2
Area of one badge = 1078 cm^2 (approximately)
Total area of cardboard = Length * Width
Total area of cardboard = 100 cm * 60 cm
Total area of cardboard = 6000 cm^2
Number of badges = Total area of cardboard / Area of one badge
Number of badges = 6000 cm^2 / 1078 cm^2
Number of badges ≈ 5.562 (approximately)
Hence, Alex can cut approximately 5 badges.
b) To calculate the area of the cardboard left over, we need to subtract the total area of the badges from the total area of the cardboard.
Area of the cardboard left over = Total area of cardboard - (Number of badges * Area of one badge)
Area of the cardboard left over = 6000 cm^2 - (5 * 1078 cm^2)
Area of the cardboard left over ≈ 6000 cm^2 - 5390 cm^2
Area of the cardboard left over ≈ 610 cm^2
Therefore, the area of the cardboard left over is approximately 610 cm^2.