An architect has allocated a rectangular space of 465ft^2 for square dining room and a 16ft wide kitchen. Find both the width of the square dining room and the length of the entire rectangular space.
w^2 + 16w = 465
w = 15
Let's x be the width of the square dining room in feet.
The area of a square is calculated by squaring the length of one of its sides. Therefore, the area of the square dining room is x^2 ft^2.
The kitchen has a width of 16 ft, so the length of the entire rectangular space is x + 16 ft.
The total area of the rectangular space is given as 465 ft^2.
Therefore, we can set up the equation:
x^2 + 16x = 465
To solve this equation, we can set it equal to zero by subtracting 465 from both sides:
x^2 + 16x - 465 = 0
Now, we can use factoring or the quadratic formula to solve for x.
Factoring:
(x - 15)(x + 31) = 0
x - 15 = 0 or x + 31 = 0
x = 15 or x = -31
Since the width cannot be negative, we discard x = -31.
Therefore, the width of the square dining room is 15 ft.
The length of the entire rectangular space is x + 16 = 15 + 16 = 31 ft.
So, the width of the square dining room is 15 ft, and the length of the entire rectangular space is 31 ft.
To solve this problem, we need to use the given information to set up two equations. Let's represent the width of the square dining room as "x" and the length of the entire rectangular space as "y".
Equation 1: Area of the square dining room
The area of a square is given by the formula A = s^2, where A is the area and s is the side length of the square. Since the architect has allocated a rectangular space of 465 ft^2 for the square dining room, we can set up the equation:
x^2 = 465
Equation 2: Area of the rectangular space
The area of a rectangle is given by the formula A = length * width. Since we know the width of the kitchen is 16 ft, and the width of the square dining room is also the width of the rectangular space, we can set up the equation:
x * y = 465 + 16y
Now, we have a system of equations to solve:
x^2 = 465
x * y = 465 + 16y
To find the value of x, we can substitute the value of y from equation 2 into equation 1:
x * (465 + 16y) = 465
Simplifying this equation further, we get:
465x + 16xy = 465
Since we are given that the width of the kitchen is 16 ft, we can further simplify the equation to:
465x + 16xy = 465 + 16 * 16
Now, we have a single equation that we can solve for x.