An architect has allocated a rectangular space of 666ft^2 for a square dining room and a 19ft wide kitchen. Find the width of the square dining room and the length of the entire rectangular space.
x^2 + 19x= 666
x^2 + 19x - 666 = 0
(x -18) (x+37)= 0
x -18 = 0
x = 18
To find the width of the square dining room, let's assume it is denoted by "w".
Given that the dining room is square, its area is simply w^2 square feet.
The architect has also allocated a 19ft wide kitchen within the rectangular space. Therefore, the remaining width after deducting the kitchen width from the total length of the rectangular space would be (w + 19) ft.
We are given that the total area of the rectangular space is 666 square feet. We can now set up an equation to solve for the width (w) of the square dining room:
w^2 + 19w = 666
Now we have a quadratic equation. Rearrange it to the form ax^2 + bx + c = 0:
w^2 + 19w - 666 = 0
We can either solve this quadratic equation by factoring or by using the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 1, b = 19, and c = -666. Substituting these values into the quadratic formula:
w = (-19 ± √(19^2 - 4(1)(-666))) / (2(1))
Simplifying further:
w = (-19 ± √(361 + 2664)) / 2
w = (-19 ± √3025) / 2
w = (-19 ± 55) / 2
Now we have two possible values for w. Solving for both:
w1 = (-19 + 55) / 2 = 36 / 2 = 18 ft
w2 = (-19 - 55) / 2 = -74 / 2 = -37 ft (not a valid solution)
Since a negative width does not make sense in this context, the width of the square dining room is 18 ft.
To find the length of the entire rectangular space, we can substitute the value of w into the equation for the remaining width:
Length = w + 19 = 18 + 19 = 37 ft
Therefore, the width of the square dining room is 18 ft and the length of the entire rectangular space is 37 ft.