The area of a rectangular garden is 140
square meters. The width is 3
meters longer than one-half
of the length.
length --- x
width ----x/2 + 3
x(x/2 + 3) = 140
x^2 /2 + 3x - 140 = 0
x^2 + 6x - 280 = 0
(x - 14)(x + 20) = 0
x = 14, we reject the x = -20
length is 14 m, width is 10 m
Let's break down the given information and determine the dimensions of the rectangular garden step-by-step:
Step 1: Let's assume the length of the rectangular garden is "L" meters.
Step 2: According to the given information, the width is 3 meters longer than one-half of the length. So, the width can be expressed as (L/2) + 3.
Step 3: We have the area formula for a rectangle: Area = Length × Width. In this case, the area is given as 140 square meters.
Step 4: Substituting the width expression into the area formula, we get the equation: 140 = L × [(L/2) + 3].
Step 5: To simplify the equation, let's distribute L to the terms inside the bracket: 140 = (L^2/2) + 3L.
Step 6: We can multiply the entire equation by 2 to eliminate the fraction: 280 = L^2 + 6L.
Step 7: Rearranging the equation to put it in standard quadratic form, we get: L^2 + 6L - 280 = 0.
Step 8: At this point, we can solve this quadratic equation using either factoring or the quadratic formula. However, this equation does not easily factor, so we will use the quadratic formula: L = (-b ± √(b^2 - 4ac)) / (2a).
Step 9: Comparing the quadratic equation with the general quadratic equation of ax^2 + bx + c = 0, we can determine the values of a, b, and c for our equation: a = 1, b = 6, c = -280.
Step 10: Substituting the values into the quadratic formula, we get: L = (-6 ± √(6^2 - 4(1)(-280))) / (2(1)).
Step 11: Simplifying the equation further, we have: L = (-6 ± √(36 + 1120)) / 2.
Step 12: Simplifying inside the square root, we get: L = (-6 ± √(1156)) / 2.
Step 13: The square root of 1156 is 34. Thus, L = (-6 ± 34) / 2.
Step 14: Considering both positive and negative solutions, we have two possible values for L: L₁ = (-6 + 34) / 2 = 28/2 = 14, and L₂ = (-6 - 34) / 2 = -40/2 = -20.
Step 15: Since the length cannot be negative, we discard the negative value. Therefore, the length of the rectangular garden is 14 meters.
Step 16: Substituting the length back into the width expression, we have: Width = (14/2) + 3 = 7 + 3 = 10 meters.
So, the length of the rectangular garden is 14 meters, and the width is 10 meters.
To solve this problem, we need to set up an equation based on the given information. Let's assume the length of the garden is represented by 'x'.
The width is stated to be 3 meters longer than one-half of the length, which can be expressed as (1/2)x + 3.
The formula for the area of a rectangle is length multiplied by width. Therefore, the equation for the area of the garden can be written as:
x * ((1/2)x + 3) = 140
We can simplify the equation:
(1/2)x^2 + 3x = 140
To solve this quadratic equation, we need to set it equal to zero and then factor or use the quadratic formula. However, in this case, I will explain how to solve it by factoring.
So, we rearrange the equation to make it equal to zero:
(1/2)x^2 + 3x - 140 = 0
To factor this equation, we need to find two numbers that multiply to -70 (the product of the leading coefficient, 1/2, and the constant term, -140), and add up to 3 (the coefficient of the 'x' term).
After trying different pairs of factors, we find that -7 and 10 are the numbers that satisfy these conditions.
Therefore, we can factor the equation as follows:
(1/2)x^2 - 7x + 10x - 140 = 0
(1/2)x(x - 7) + 10(x - 7) = 0
(x - 7)(1/2x + 10) = 0
Now we have two separate factors, x - 7 = 0 and 1/2x + 10 = 0, which are both set to zero.
Solving for x in the first factor, we have:
x - 7 = 0
x = 7
Solving for x in the second factor, we have:
1/2x + 10 = 0
1/2x = -10
x = -20
Since the dimensions of a garden cannot be negative, we discard the solution x = -20.
Therefore, the length of the rectangular garden is 7 meters.
And according to the given information, the width is 3 meters longer than one-half of the length. Therefore, the width would be:
(1/2)(7) + 3 = 3.5 + 3 = 6.5 meters
So, the length of the garden is 7 meters and the width is 6.5 meters.