The floor of a one-story building is 14 feet longer than it is wide (see figure). The building has 1472 square feet of floor space.
a) Write a quadratic equation for the area of the floor in terms of w.
(b) Find the length l and width w of the floor.
a) Well, let's start by defining our variables. Let's say w represents the width of the floor.
Now, to find the length of the floor, we know that it is 14 feet longer than the width. So, the length can be represented as (w + 14).
The overall area of the floor can be calculated by multiplying the length and the width, which gives us:
Area = Length * Width
Area = (w + 14) * w
b) To find the dimensions of the floor, we need to solve the quadratic equation. But before we do that, we are given that the area of the floor is 1472 square feet.
So, we can set up the equation:
(w + 14) * w = 1472
Expanding this equation, we get:
w^2 + 14w = 1472
Rearranging the terms and setting the equation to zero, we have:
w^2 + 14w - 1472 = 0
Now we can solve this quadratic equation to find the values of w, which will give us the width of the floor.
To solve this problem, let's denote the width of the floor as "w" and the length as "l".
a) To write a quadratic equation for the area of the floor in terms of w, we can start by expressing the area of the floor as the product of its length and width. Since the length is 14 feet longer than the width, we can write the equation as follows:
Area = length * width
Since the length is 14 feet longer than the width, we can express the length as w + 14. Therefore, the equation becomes:
Area = (w + 14) * w
Expanding this equation, we get:
Area = w^2 + 14w
So, the quadratic equation for the area of the floor in terms of w is:
Area = w^2 + 14w
b) To find the length l and width w of the floor, we need to solve the quadratic equation and find the values of w that satisfy the given conditions. Let's set the area equal to 1472 square feet:
Area = w^2 + 14w = 1472
Rearranging this equation, we get:
w^2 + 14w - 1472 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 14, and c = -1472. Substituting these values into the quadratic formula, we have:
w = (-(14) ± √((14)^2 - 4(1)(-1472))) / (2 * 1)
Simplifying this equation further:
w = (-14 ± √(196 + 5888)) / 2
w = (-14 ± √(6084)) / 2
w = (-14 ± 78) / 2
This gives us two possible values for w:
w = (-14 + 78) / 2 or w = (-14 - 78) / 2
w = 64 / 2 or w = -92 / 2
w = 32 or w = -46
Since width cannot be negative, the width w = 32 feet.
To find the length l, we can substitute the value of w into the equation for the length:
l = w + 14
l = 32 + 14
l = 46
Therefore, the length l of the floor is 46 feet and the width w is 32 feet.
To solve this problem, we can use the formula for the area of a rectangle, which is length multiplied by width. Let's take the width of the floor as 'w'. In that case, the length of the floor will be 'w + 14' since it is 14 feet longer than the width.
a) Write a quadratic equation for the area of the floor in terms of w.
The equation for the area (A) of the floor is A = length * width. Substituting the values, we have:
A = (w + 14) * w
Expanding the equation, we get:
A = w^2 + 14w
So the quadratic equation for the area of the floor in terms of w is A = w^2 + 14w.
b) Find the length l and width w of the floor.
The problem gives us the information that the floor has an area of 1472 square feet. So, we can set up the equation A = 1472 and solve for w.
w^2 + 14w = 1472
To solve this equation, we can set it equal to zero and factor or use the quadratic formula. Let's use factoring:
w^2 + 14w - 1472 = 0
Factoring, we find:
(w + 56)(w - 26) = 0
This gives us two possible solutions for w: w = -56 and w = 26.
Since the width of the floor cannot be negative, we discard the negative solution. Therefore, the width of the floor is w = 26 feet.
To find the length (l) of the floor, we substitute the value of w into the equation for the length of the floor: l = w + 14.
l = 26 + 14
l = 40
Therefore, the length of the floor is l = 40 feet and the width is w = 26 feet.
w (w+14) = 1472
w^2 + 14 w - 1472 = 0
w = [-14 +/- sqrt(196+5888) ]/2
w = [-14 +/- 78]/2
want positive w so
w = 32
then l = 32+14 = 46