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.
W = H + 4
L = W + 8 = H + 4 + 8 = H + 12
A = 2H (W + L) = 2 H (2H + 16)
512 = 4 H² + 32 H
128 = H² + 8 H
0 = H² + 8 H - 128
solve for H, then substitute back to find W and L
To find the dimensions of the room, we can set up a system of equations based on the given information.
Let's assume the height of the room is h meters, the width is w meters, and the length is l meters.
Based on the first statement, we know that the width is 4 meters less than the length, so we can write the equation:
w = l - 4 ...(1)
According to the second statement, the length is 8 meters longer than the width, so we can write the equation:
l = w + 8 ...(2)
Now, let's calculate the area of the room's walls. The formula for the surface area (SA) of a rectangular room is:
SA = 2lw + 2lh ...(3)
Given that the total area of the walls is 512 square meters, we have:
2lw + 2lh = 512 ...(4)
Let's substitute equations (1) and (2) into equation (4):
2(w)(w + 8) + 2(w)(w - 4) = 512
Simplifying the equation:
2w^2 + 16w + 2w^2 - 8w = 512
Combining like terms:
4w^2 + 8w - 512 = 0
Simplifying further:
w^2 + 2w - 128 = 0
Now, we can solve this quadratic equation using factorization or the quadratic formula. Factoring the equation, we get:
(w - 8)(w + 16) = 0
This gives us two possible values for w: w = 8 or w = -16.
Since width cannot be negative, we discard w = -16 and consider w = 8 as the valid solution.
Now, substituting w = 8 into equation (1), we can find the length:
l = w + 8 = 8 + 8 = 16
Finally, substituting w = 8 and l = 16 into equation (2), we can find the height:
h = l - 4 = 16 - 4 = 12
So, the dimensions of the room are:
Width = 8 meters
Length = 16 meters
Height = 12 meters