Alex wants to cut out as many circular badges as possible from this rectangular sheet of cardboard. the diameter of each badge is 14 cm.
a) how many badges can he cut?
b) what is the area of the cardboard left over?
a) To determine how many badges can be cut from the rectangular sheet of cardboard, we need to first find the dimensions of the sheet. Let's label the length as L and the width as W.
We know that the diameter of each badge is 14 cm, which means the radius is 7 cm. To cut out a badge, we need a square of cardboard that is at least as big as the circle (so the diagonal of the square is equal to the diameter of the circle). Therefore, the side length of each square needed to cut out a badge is 14 cm (since 14 cm is the diameter of the circle).
We can fit as many squares as possible into the length and width of the cardboard. In other words, the number of squares that fit along the length of the cardboard is equal to L/14, rounded down to the nearest whole number. Similarly, the number of squares that fit along the width of the cardboard is equal to W/14, rounded down to the nearest whole number.
Therefore, the total number of badges that can be cut is:
L/14 x W/14
b) The area of the cardboard left over is equal to the total area of the rectangle minus the total area of all the badges that were cut out.
The total area of the cardboard is:
L x W
The total area of each badge is:
πr^2 = π(7 cm)^2 ≈ 153.94 cm^2
Therefore, the total area of all the badges that were cut out is:
[L/14 x W/14] x 153.94
And the area of the cardboard left over is:
L x W - [L/14 x W/14] x 153.94
To find out how many badges Alex can cut from the rectangular sheet, we need to determine the maximum number of circles that can fit within the rectangle.
To calculate the number of badges Alex can cut from the rectangular sheet, we need to divide the length and width of the rectangle by the diameter of each badge.
a) Calculation for the number of badges:
Length of the rectangular sheet = ?
Width of the rectangular sheet = ?
Diameter of each badge = 14 cm
Since the length and width of the rectangular sheet are not provided in the question, we are unable to calculate the exact number of badges that can be cut. Could you please provide the measurements of the length and width of the rectangular sheet?
b) Once we have the measurements of the length and width of the rectangular sheet, we can calculate the area of the cardboard left over. The area of the cardboard left over will be the total area of the rectangular sheet minus the total area of all the badges that are cut.
Let's assume the length of the rectangular sheet is L and the width is W. Then:
Area of the rectangular sheet = L × W
Area of each badge = π × (diameter/2)^2 = π × (14/2)^2
The total area of all the badges that can be cut is the area of each badge multiplied by the number of badges, which is the number we will calculate in part a).
Area of cardboard left over = Area of the rectangular sheet - Total area of all the badges
Please provide the measurements of the length and width of the rectangular sheet so that we can proceed with the calculations.