A square has X cm each side. If each side of the square is increased by 4 cm, the are us increased by 392 cm^2 . Calculate x.
Let's start by using the formula for the area of a square, which is A = s^2, where A is the area and s is the length of a side.
If the length of each side of the square is increased by 4 cm, then the new length of a side is x + 4 cm. The new area of the square is (x + 4)^2.
We know that the area is increased by 392 cm^2, so we can set up an equation:
(x + 4)^2 - x^2 = 392
Expanding the left side of the equation, we get:
x^2 + 8x + 16 - x^2 = 392
Simplifying, we get:
8x + 16 = 392
Subtracting 16 from both sides, we get:
8x = 376
Dividing both sides by 8, we get:
x = 47
Therefore, the length of each side of the original square is 47 cm.
To solve this problem, let's break it down step by step:
1. Let's assume the length of each side of the square is X cm.
2. If each side is increased by 4 cm, then the new length of each side would be X + 4 cm.
3. The area of the original square is X * X = X^2 cm^2.
4. The area of the new square is (X + 4) * (X + 4) = (X + 4)^2 cm^2.
5. We are given that the increase in area is 392 cm^2, so we can set up the equation:
(X + 4)^2 - X^2 = 392
6. Expanding the equation, we get:
X^2 + 8X + 16 - X^2 = 392
7. Simplifying the equation, we have:
8X + 16 = 392
8. Subtracting 16 from both sides, we get:
8X = 376
9. Finally, dividing both sides by 8, we find:
X = 47
Therefore, the length of each side of the square is 47 cm.
Let's break down this problem step by step:
Step 1: Determine the initial area of the square
Since the side length of the square is X cm, the initial area can be calculated using the formula: area = side length squared. Therefore, the initial area is X^2 cm^2.
Step 2: Determine the new side length of the square
If each side of the square is increased by 4 cm, the new side length of the square will be X + 4 cm.
Step 3: Determine the new area of the square
Using the same formula for area (area = side length squared), the new area of the square can be calculated as (X + 4)^2 cm^2.
Step 4: Calculate the difference between the new and initial areas
The problem states that the area is increased by 392 cm^2, so we can subtract the initial area from the new area to find the difference: (X + 4)^2 cm^2 - X^2 cm^2 = 392 cm^2.
Step 5: Solve the quadratic equation
Expanding the equation (X + 4)^2 - X^2 = 392, we get X^2 + 8X + 16 - X^2 = 392.
Simplifying further, we are left with 8X + 16 = 392.
Step 6: Solve for X
To isolate X, we subtract 16 from both sides of the equation: 8X = 392 - 16, which gives us 8X = 376.
Finally, we divide both sides of the equation by 8: X = 376/8.
Step 7: Calculate X
Simplifying the equation further, we have X = 47.
Therefore, X = 47 cm.
So, the side length of the square is 47 cm.