Algebra Word Problems:
1. The length of the floor of a one-storey building is 14 feet longer than its width. The building has 1,632 square feet of floor space. Write a quadratic equation for the area of the floor in terms of w. Find the length and width of the floor.
2. A gardener has 100 meters of fencing to enclose two adjacent rectangular gardens. The gardener wants the enclosed area to be 400 square meters. What dimensions should the gardener use to obtain this area?
1.
width --- w
length -- w+14
w(w+14) = 1632
w^2 + 14w = 1632
w^2 + 14w + 49 = 1632 + 49
I completed the square
(w+7)^2 = 1681
w+7 =±41
w = -7 ± 41 = 34 or a negative
the width is 34 ft and the length is 48 ft
or
it factors to
(x-34)(x+48) = 0
x = 34 or x = -48
2.
width ---- x
lengty --- y
2x + 2y = 100
x+y = 50
y = 50-x
area = 400
xy = 400
x(50-x) = 400
50x - x^2 - 400 = 0
x^2 - 50x + 400 = 0
x^2 - 50x = -400
x^2 - 50x + 625 = -400 + 625
(x-25)^2 = 225
x-25 = ± 15
x = 25 ± 15
x = 40 or x = 10
if x = 40, y = 10
if x = 10, y = 40
the garden should be 10 m by 40 m
1. To write a quadratic equation for the area of the floor in terms of w, we need to find the length and width of the floor.
Let's assume:
Width of the floor = w feet
Length of the floor = w + 14 feet
The area of the floor is given by the formula: Area = Length * Width
Substituting the values, we have:
Area = (w + 14) * w = 1,632
We can rewrite this equation as a quadratic equation by expanding it:
w^2 + 14w = 1,632
The equation in quadratic form is:
w^2 + 14w - 1,632 = 0
To find the length and width of the floor, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
2. To find the dimensions of the rectangular gardens, we need to consider that the enclosed area is 400 square meters and the total length of the fencing is 100 meters.
Let's assume:
Length of the rectangular garden = L meters
Width of the rectangular garden = W meters
The area of the garden is given by the formula: Area = Length * Width
Substituting the values, we have:
Area = L * W = 400
The total length of fencing is given by the formula: Perimeter = 2 * (Length + Width)
Substituting the values, we have:
Perimeter = 2 * (L + W) = 100
We can solve this system of equations to find the dimensions of the rectangular gardens. Here's one way to do it:
From the first equation, we can isolate L:
L = 400 / W
Substituting this value of L into the second equation:
2 * ((400 / W) + W) = 100
Simplifying the equation:
800 / W + 2W = 100
800 + 2W^2 = 100W
2W^2 - 100W + 800 = 0
Now, we have a quadratic equation in terms of W. We can solve this equation using factoring, completing the square, or the quadratic formula to find the width and length of the rectangular gardens.
To solve these algebra word problems, we need to represent the given information mathematically and then solve the resulting equations. Let's go step by step:
1. The length of the floor of a one-story building is 14 feet longer than its width. The building has 1,632 square feet of floor space. Write a quadratic equation for the area of the floor in terms of w. Find the length and width of the floor.
Step 1: Represent the width of the floor:
Let's assume the width of the floor is w feet.
Step 2: Represent the length of the floor:
According to the problem, the length is 14 feet longer than the width. So, the length of the floor would be w + 14 feet.
Step 3: Write the quadratic equation:
The area of a rectangle is given by the formula A = length * width.
In this case, the area of the floor is given as 1,632 square feet. So we can write the equation as follows:
A = (w + 14) * w = 1632
Step 4: Solve the quadratic equation:
To find the length and width, we need to solve the quadratic equation (w + 14) * w = 1632.
Expanding the equation, we get:
w^2 + 14w = 1632
Rearranging the equation:
w^2 + 14w - 1632 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
2. A gardener has 100 meters of fencing to enclose two adjacent rectangular gardens. The gardener wants the enclosed area to be 400 square meters. What dimensions should the gardener use to obtain this area?
Step 1: Let's assume one side of the rectangular garden to be x meters. Then the adjacent side would be (100 - 2x) meters because the gardener uses fencing on all sides.
Step 2: The area of the rectangular garden is given by the formula A = length * width. In this case, it should be 400 square meters. So we can write the equation as:
A = x(100 - 2x) = 400.
Step 3: Solve the quadratic equation:
To find the dimensions of the rectangular garden, we need to solve the quadratic equation x(100 - 2x) = 400.
Expanding and rearranging the equation will result in a quadratic equation:
-2x^2 + 100x - 400 = 0.
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.