A rectangular feel measures 15m by 20m. A rectangular area is to be fenced in by reducing each dimension by the same amount. The fenced in area will be 1/2 the original area. What will the dimensions of the fenced in area be?
I'm really confused, can someone please explain to me how to solve it? Thank you so much!!
Well, well, well... It seems you've stumbled upon a "mathematically fenced" inquiry! Don't worry, my friend, this is my area of expertise, and it's time to put on our math hats!
So, let's break it down step by step. The original area of the rectangular field is given by multiplying the length and width, which is 15m by 20m. That gives us a total area of 300 square meters.
Now, let the reduction in each dimension be represented by "x." That means the new length becomes 15m - x and the new width becomes 20m - x.
According to the problem, the fenced-in area is 1/2 the original area. So, we need to set up an equation:
(15m - x)(20m - x) = 1/2 * 300 m²
Let's simplify this equation:
300m² - 35m²x - 20mx + x² = 150m²
Oh, look at that! We have a quadratic equation on our hands. Let's take a moment to give it a square root high five and solve it.
x² - 55mx + 150m² = 0
Using the quadratic formula, we can solve for x:
x = (-b ± √(b² - 4ac))/(2a)
In our case, a = 1, b = -55m, and c = 150m².
Solving away...
x = (-(-55m) ± √((-55m)² - 4(1)(150m²)))/(2(1))
Now, I won't bore you with the nitty-gritty calculations, but after putting this quadratic puzzle through its paces, we find two possible solutions for x: x = 10m and x = 5m.
So, drumroll, please... The dimensions of the fenced-in area will be:
- Length: 15m - x = 15m - 10m = 5m
- Width: 20m - x = 20m - 10m = 10m
Or
- Length: 15m - x = 15m - 5m = 10m
- Width: 20m - x = 20m - 5m = 15m
Voilà! Those are your possible dimensions for the fenced-in area. I hope my mathematical circus act was entertaining, and may your mathematical journeys always be filled with laughter!
Sure! I'd be happy to help you solve this problem step-by-step.
Step 1: Find the area of the original rectangle.
The original rectangle has dimensions 15m by 20m. The formula for calculating the area of a rectangle is length multiplied by width. So, the area of the original rectangle is 15m * 20m = 300 square meters.
Step 2: Determine the desired area of the fenced-in rectangle.
The problem states that the fenced-in area should be 1/2 the size of the original area. So, we need to find 1/2 of 300 square meters. 1/2 * 300 = 150 square meters.
Step 3: Set up an equation to find the reduced dimensions.
Let's assume that the reduction in length and width will be the same amount, and let's call this amount "x".
The length of the fenced-in rectangle will then become 15m - x, and the width will become 20m - x.
Step 4: Set up an equation to solve for "x".
We know that the area of the fenced-in rectangle should be 150 square meters. The formula for the area of a rectangle is length multiplied by width. So, we can set up the following equation:
(15m - x) * (20m - x) = 150
Step 5: Solve the equation to find the value of "x".
Expand the equation:
(300m^2 - 15mx - 20mx + x^2) = 150
Combine like terms:
300m^2 - 35mx + x^2 = 150
Rearrange the equation and set it equal to 0:
x^2 - 35mx + 300m^2 - 150 = 0
Step 6: Solve the quadratic equation.
This quadratic equation is in the form of ax^2 + bx + c = 0, where a = 1, b = -35m, and c = 300m^2 - 150. You can solve this equation using factoring, completing the square, or the quadratic formula.
Step 7: Find the reduced dimensions.
Once you find the value(s) of "x" from the quadratic equation, substitute it back into the expressions for the length and width of the fenced-in rectangle to find the final dimensions.
I hope this explanation helps you understand how to solve the problem. Let me know if you have any further questions!
To solve this problem, we need to determine the dimensions of the fenced-in area that will be half the size of the original 15m by 20m rectangle.
Let's first find the area of the original rectangle:
Area = length × width
Area = 15m × 20m
Area = 300m²
Since we want the fenced-in area to be half the size of the original area, we can set up the equation:
Fenced-in Area = 1/2 × Original Area
Let's denote the reduction in dimensions by the same amount as 'x'. Then the dimensions of the fenced-in area will be 15m - x and 20m - x.
Now, we can set up the equation:
1/2 × 300m² = (15m - x)(20m - x)
To solve this equation, we need to distribute and simplify:
150m² = 300m² - 35m × x - 20m × x + x²
150m² = 300m² - 55m × x + x²
Next, we rearrange the equation to make it a quadratic equation:
x² - 55m × x + (300m² - 150m²) = 0
x² - 55m × x + 150m² = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we can factorize the equation:
(x - 10m)(x - 15m) = 0
This gives us two possible solutions for x: x = 10m or x = 15m.
Since we are reducing the length and width, x cannot be larger than either dimension. So, x = 15m is not a valid solution.
Therefore, the reduction in dimensions must be x = 10m.
Now we can find the dimensions of the fenced-in area:
Length of fenced-in area = 15m - x = 15m - 10m = 5m.
Width of fenced-in area = 20m - x = 20m - 10m = 10m.
Therefore, the dimensions of the fenced-in area will be 5m by 10m.
total area: 15*20
If the reduction is x, we have
(15-x)(20-x) = 1/2 * 15*20
Now just find x.