an architect has allocated a rectanguoar space of 324ft^2 for a square dinging room and a 12ft wide kitchen. find both width of the dining room and the length of the entire rectangular space.
324=12*27
So, if the width of the area is 12, so is the width of the dining room.
I assume you can figure out the total length now.
To find the width of the dining room and the length of the entire rectangular space, we can set up an equation using the given information.
Let's assume that the width of the dining room is 'x' feet. Since the dining room is square, the length of the dining room is also 'x' feet.
The area of a square is calculated by multiplying its side length by itself. Therefore, the area of the square dining room is x * x = x^2 square feet.
The kitchen is 12 feet wide, so its length is (x + 12) feet.
The total area of the rectangular space is given as 324 square feet. So, the equation becomes:
x^2 + 12x = 324
Now, we need to solve this quadratic equation to find the value of 'x' (the width of the dining room).
Step 1: Rearrange the equation to make it equal to zero:
x^2 + 12x - 324 = 0
Step 2: Solve the equation using factoring, completing the square, or using the quadratic formula. In this case, let's use factoring to solve the quadratic equation.
Factoring the quadratic equation, we need to find two numbers that multiply to -324 and add up to 12. After careful calculation, we find that the numbers are 18 and -18.
So, the equation factors to:
(x + 18)(x - 18) = 0
Setting each factor equal to zero:
x + 18 = 0 or x - 18 = 0
Solving for 'x':
x = -18 or x = 18
Since the width cannot be a negative value, we discard x = -18 as an extraneous solution.
Therefore, the width of the dining room (and the length of the dining room) is x = 18 feet.
To find the length of the entire rectangular space, we substitute the value of 'x' into the expression x + 12:
Length of entire rectangular space = 18 + 12 = 30 feet.
Hence, the width of the dining room is 18 feet, and the length of the entire rectangular space is 30 feet.