the width of a rectangular room is 4m wider tan its height and the length is 8m wider than the width .if the total area of the wall is 512m square, find the length of the room.

height is x,

width = x+4
length = w+8 = x+12

area of walls is 2x(x+4) + 2x(x+12) = 2x(2x+16) = 4x(x+8)

since 512 = 2^9, I guess 4*8*16=512 so x=8

50 feet fencing to build a rectangular pen

50 feet fencing to build a rectangular pen length width and area probelem box solve problem

To solve this problem, we can break it down into smaller steps.

Step 1: Assign variables.
Let's assign variables to the unknowns in the problem. Let:
- Height of the room = h
- Width of the room = w
- Length of the room = l

Step 2: Write equations based on the given information.
From the problem, we can derive two equations:
- The width of the room is 4m wider than its height: w = h + 4
- The length of the room is 8m wider than the width: l = w + 8

Step 3: Form an equation using the total area.
The total area of the wall is given as 512m². The area of a rectangular wall is calculated by multiplying the length and the width. So, we can write another equation:
- Area of the wall = Length × Width
- 512 = l × w

Step 4: Substitution.
Now that we have three equations, we can substitute the values from one equation into another. Let's substitute the values of l and w from Step 2 into the equation from Step 3:
- 512 = (w + 8) × w

Step 5: Simplify and solve.
Now, expand the equation:
- 512 = w^2 + 8w

Rearrange the equation into a quadratic form:
- w^2 + 8w - 512 = 0

This equation cannot be easily factored, so we can use the quadratic formula to solve for w. The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = 8, and c = -512. Applying the quadratic formula to solve for w:
- w = (-8 ± √(8^2 - 4(1)(-512))) / (2(1))

Simplifying and calculating the value of w:
- w = (-8 ± √(64 + 2048)) / 2
- w = (-8 ± √2112) / 2

Now, calculate both values of w:
- w ≈ (-8 + 45.95) / 2 ≈ 37.95 / 2 ≈ 18.975
- w ≈ (-8 - 45.95) / 2 ≈ -53.95 / 2 ≈ -26.975 (since width cannot be negative, we ignore this value)

Step 6: Find the length.
Using the value of w, we can find the length of the room by substituting it into one of the equations in Step 2. Let's use the equation l = w + 8:
- l = 18.975 + 8 = 26.975

Therefore, the length of the room is approximately 26.975 meters.