A rectangular field has a width of x meters.
The length of the field is 25m greater than twice the width of the field.
The area of the field is 450m²
Work out the length of the field.
Someone please help me
y = 2 x + 25
x y = 450
so x = 450/y
y = 2 (450/y) + 25
y^2 = 900 + 25 y
y^2 - 25 y - 900 = 0
(y-45)(y+20) = 0
y = 45
To find the length of the field, we can use the given information and utilize the formula for the area of a rectangle.
1. Let's start by defining the width of the field as x meters.
2. According to the problem, the length of the field is 25 meters greater than twice the width. We can express this as (2x + 25) meters.
3. Now, we can use the formula for the area of a rectangle: Area = Length * Width.
Given that the area of the field is 450m², we can set up the equation:
450 = (2x + 25) * x
4. Simplify the equation:
450 = 2x² + 25x
5. Rearrange the equation to set it equal to zero:
2x² + 25x - 450 = 0
6. Now, we need to solve this quadratic equation. We can factor it or use the quadratic formula:
Factoring:
(2x - 15)(x + 30) = 0
Setting each factor equal to zero gives us two possible solutions:
2x - 15 = 0 or x + 30 = 0
Solving each equation:
2x = 15 x = -30
x = 15/2
Since the width of the field cannot be negative, we discard -30 as a valid solution. Therefore, x = 15/2.
7. Now that we know the width of the field is x = 15/2 = 7.5 meters, we can substitute this value back into the expression for the length of the field:
Length = 2x + 25
Length = 2(7.5) + 25
Length = 15 + 25
Length = 40 meters
So, the length of the field is 40 meters.
To work out the length of the field, we will break down the problem and solve it step by step.
1. First, let's assign variables to the given information:
- Width of the field = x meters
- Length of the field = ? (we need to find this)
- The length of the field is 25 meters greater than twice the width, which can be expressed as: 2x + 25.
2. The area of a rectangle is given by the formula: Area = Length x Width.
In this case, the area of the field is given as 450m². So, we can write the equation:
450 = Length x Width.
3. Substitute the given values into the equation:
450 = (2x + 25) x x.
4. Simplify the equation:
450 = (2x² + 25x).
5. Rearrange the equation to make it easier to solve:
2x² + 25x - 450 = 0.
6. Now, we can solve this quadratic equation. There are a few ways to do this, but let's use factoring in this case.
- Begin by finding two numbers that multiply to give the product of the coefficient of x² (2) and the constant term (-450). The numbers that satisfy this condition are -50 and 9.
- Rewrite the middle term (25x) using these two numbers:
2x² - 50x + 9x - 450 = 0.
- Group the terms:
(2x² - 50x) + (9x - 450) = 0.
- Factor out the common factors from each group:
2x(x - 25) + 9(x - 25) = 0.
- Factor out the common term (x - 25):
(2x + 9)(x - 25) = 0.
7. Set each factor equal to zero and solve for x:
- 2x + 9 = 0 --> 2x = -9 --> x = -9/2 (we discard this solution as it gives a negative width, which is not possible).
- x - 25 = 0 --> x = 25.
8. The width of the field (x) is 25 meters. Now, substitute this value into the expression for the length of the field:
Length = 2x + 25 = 2(25) + 25 = 50 + 25 = 75 meters.
Therefore, the length of the field is 75 meters.