A rectangular room is 4 meters wider than its high and it is 8 meters longer than its wide. the total Ares of the wall is 512 square meters. Find the dimensions of the room.

height: h

width: h+4
length: w+8 = h+12

h(h+4)(h+12) = 512

all the factors of 512 are powers of 2, so we have

2^9 = 512
2+3+4 = 9
2^2 * 2^3 * 2^4 = 512
4*8*16 = 512

To access answer to the question

To find the dimensions of the room, let's assign variables to the unknown values.

Let's say the width of the room is "w" meters.
Since the room is 4 meters wider than its height, the height would be "w + 4" meters.

Given that the room is 8 meters longer than its width, the length would be "w + 8" meters.

Now, let's break down what we know and set up an equation:
The total area of the walls is 512 square meters.
The area of each wall can be calculated by multiplying the width by the height.
In a rectangular room, there are two walls with the same dimensions and two other walls with the same dimensions.

So, the equation would be:
2(w + 8)(w) + 2(w + 4)(w) = 512

To solve this equation, we can simplify it step by step:

2(w + 8)(w) + 2(w + 4)(w) = 512
2w² + 16w + 2w² + 8w + 4w + 16 = 512
4w² + 30w + 16 = 512
4w² + 30w - 496 = 0

Now we can solve this quadratic equation by factoring or by using the quadratic formula. Factoring may not yield whole numbers, so let's use the quadratic formula:

w = (-b ± √(b² - 4ac)) / (2a)

In our equation, a = 4, b = 30, and c = -496.

w = (-30 ± √(30² - 4 * 4 * -496)) / (2 * 4)
w = (-30 ± √(900 + 7936)) / 8
w = (-30 ± √8836) / 8
w = (-30 ± 94) / 8

Now we have two possible values for w:
w₁ = (-30 + 94) / 8 = 64 / 8 = 8

w₂ = (-30 - 94) / 8 = -124 / 8 = -15.5 (negative width doesn't make sense)

Therefore, the width of the room is 8 meters.

Now, we can find the height:
height = width + 4
height = 8 + 4 = 12 meters

And, we can find the length:
length = width + 8
length = 8 + 8 = 16 meters

So, the dimensions of the room are:
Width = 8 meters
Height = 12 meters
Length = 16 meters