An architect has allocated a rectangular space of 264ft^2 for a square dining room and a 10ft wide kitchen. What is both the width of the square dining room and the length of entire space.

Let's assume the width of the square dining room is "x" feet.

Since the area of a square is given by the formula A = side × side, we can write the equation: x × x = 264.

Simplifying the equation, we have x^2 = 264.

To find the value of x, we take the square root of both sides: √(x^2) = √264.

So, we have x = √264.

Next, we need to find the length of the entire space. Since the dining room is a square, both the width and length of the entire space would be x + 10.

Therefore, the width of the square dining room is √264 feet, and the length of the entire space is (√264 + 10) feet.

To find the width of the square dining room, we first need to calculate the area of the kitchen.

Given that the kitchen has a width of 10ft and the entire space is rectangular, we know that the kitchen's length is the same as the length of the entire space.

Let's assume the width of the square dining room is x.

The area of the kitchen is then given by:
Kitchen area = length of entire space × width of kitchen
= x × 10ft

Now, the remaining area of the rectangular space after allocating the kitchen is:
Remaining area = Total area - Kitchen area
= 264ft^2 - (x × 10ft)

Since the remaining area is allocated for the square dining room, we can set it equal to the area of a square, which is the square of its side length:
Remaining area = Side length of square dining room × Side length of square dining room
= x^2

Setting up an equation for the remaining area, we have:
264ft^2 - (x × 10ft) = x^2

Now, we can solve this quadratic equation to find the width of the square dining room (x).
To simplify the equation, let's expand and rearrange the terms:
x^2 + 10x - 264 = 0

We can factor this quadratic equation or use the quadratic formula to find the possible values for x. Then, we can select the positive value for x since dimensions cannot be negative.

Once we find the value of the width of the square dining room (x), we can use it to find the length of the entire space by substituting it back into the equation for the area of the kitchen (length of entire space = x × 10ft).

width = x

length = y
kitchen is 10 by x
dining room is square so (y-10) = x

kitchen area = 10 x
dining area = x^2
so
10 x + x^2 = 264
x^2 + 10 x - 264 = 0
(x-12)(x+22) = 0
x = 12
y = x+10 = 22

so
living room is 12 by 12
length is 22