an architect has allocated a rectangular space of 325ft^2 for a square dining room and a 12ft wide kitchen. find bothy the width of the square dining room and the length of the entire rectangular space.
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To find the dimensions of the square dining room and the length of the entire rectangular space, we can use the concept of area.
Let's assume the width of the square dining room is "x" feet. Since it is a square, the length will also be "x" feet.
The area of the square dining room is given by x * x = x^2 square feet.
Now, the remaining rectangular space is 325 ft^2 minus the area of the square dining room. So, the area of the kitchen and remaining space is 325 - x^2 square feet.
We are also given that the kitchen is 12 feet wide. Therefore, the length of the kitchen and remaining space is (325 - x^2) / 12 feet.
Since we have the width and length of the rectangular space, we can form an equation:
x * ((325 - x^2) / 12) = 325
Now, let's solve this equation to find the values of x and ((325 - x^2) / 12), which represents the width and length of the rectangular space, respectively.
Multiplying both sides of the equation by 12:
12x * ((325 - x^2) / 12) = 325 * 12
x * (325 - x^2) = 3900
Expanding the equation:
325x - x^3 = 3900
Rearranging the equation:
x^3 - 325x + 3900 = 0
To find the solutions of this cubic equation, we can use numerical methods or a graphing calculator.
Using numerical methods or a calculator, we find that the approximate solution for x is x ≈ 13.65 feet.
So, the width of the square dining room is approximately 13.65 feet, and the length of the entire rectangular space is (325 - x^2) / 12 ≈ 15.42 feet.