There is a circular table and it has an area of 1256 square inches. The table opens in the middle to allow additional boards to be added. Each board is 12 inches squared, fits across the diameter of the table, and is rectangular in shape.
How many boards must be added to increase the area to 2696 square inches? Use 3.14 fro pi.
You don't need to use pi. Let x = number of boards.
1256 + 12x = 2696
Solve for x.
To solve this problem, we need to follow these steps:
1. Find the radius of the circular table.
2. Calculate the current area of the table.
3. Calculate the area of each board.
4. Determine how many boards are needed to increase the table's area to the desired value.
Let's go through each step in detail:
Step 1: Find the radius of the circular table.
Since we know the area of the table and the formula to calculate the area of a circle (A = πr^2), we can rearrange the formula to find the radius (r).
Given that the area of the table is 1256 square inches, we have:
1256 = 3.14 * r^2
Divide both sides of the equation by 3.14:
r^2 = 1256 / 3.14
r^2 = 400
Taking the square root of both sides, we find:
r = 20 inches.
Step 2: Calculate the current area of the table.
Since we know the radius from Step 1, we can use the formula for the area of a circle (A = πr^2) to calculate the current area.
Given that the radius is 20 inches, we have:
Current area = π * (20)^2 = 400π square inches.
Step 3: Calculate the area of each board.
Given that each board is rectangular and fits across the diameter of the table, it will split the circular table into two halves. Therefore, the length of the board is equal to the diameter of the table. The area of each board can be calculated by multiplying the length of the board by its width, which is given as 12 inches.
Length of board = Diameter of table = 2 * radius = 2 * 20 = 40 inches.
Area of each board = Length * Width = 40 * 12 = 480 square inches.
Step 4: Determine how many boards are needed to increase the table's area to the desired value.
The desired area of the table is given as 2696 square inches.
We need to find the difference between the desired area and the current area of the table:
Desired increase in area = Desired area - Current area
= 2696 - 400π square inches.
To determine how many boards are needed, divide the desired increase in area by the area of each board:
Number of boards needed = Desired increase in area / Area of each board
= (2696 - 400π) / 480.
So, to increase the area of the circular table to 2696 square inches using 12-square-inch boards, you need to add (2696 - 400π) / 480 boards.