A farmer wants to divide his 2700 km2 of land into two separate pastures for his new horses by building a fence that will enclose both pastures. He wants one pasture to be twice the size as the other. What should the dimensions of the larger pasture be if he wants to use the least amount of fencing
All kinds of questions since you don't describe the enclosures.
Do the two fields have the same width?
Are there two disjointed fields with the same width and one length twice the other?
If they are joined by a common width, is that common width included in the total perimeter?
I don't have any additional information except the answer 60km×30km
it's multiple choice question
I used A=xy for the small pasture and A=2xy for the large pasture and added together but can't get the same answer 😥
just a guess, it seems like you are asked about only dimensions of the big one, isnt that just 2/3 of 2700 km2. that is 1800km2 which = 60 km x 20km.
Well, to minimize the amount of fencing, the farmer should go for a circular shape for the larger pasture. Why? Because a circle has the maximum area with the least amount of perimeter. So, let's calculate!
Let's assume the larger pasture has radius 'r' and the smaller one has radius '2r' (as it should be half the size). The area of a circle is given by the formula A = πr².
So, the larger pasture has an area of A₁ = πr², and the smaller one has an area of A₂ = π(2r)² = 4πr².
We know that the total area of the two pastures is 2700 km², so:
A₁ + A₂ = 2700 km²
πr² + 4πr² = 2700 km²
5πr² = 2700 km²
r² = 2700 km² / 5π
Now, to find the dimensions, we need the radius 'r'. So, let's calculate that!
r = √(2700 km² / 5π)
r ≈ 19.26 km
Therefore, the radius of the larger pasture is approximately 19.26 km. Since the radius is the same as the diameter in a circle, the dimensions of the larger pasture would be about 38.52 km in diameter. That's a whole lot of space for those new horses!
To determine the dimensions of the larger pasture, we need to follow these steps:
Step 1: Let's assign variables to the dimensions of the smaller pasture. Let's say the width of the smaller pasture is x km and the length is y km.
Step 2: Since the larger pasture is twice the size of the smaller one, the width and length of the larger pasture would be 2x km and 2y km, respectively.
Step 3: The area of the smaller pasture is x * y km².
Step 4: The area of the larger pasture is 2x * 2y km², which simplifies to 4xy km².
Step 5: According to the question, the area of both pastures combined should be 2700 km². So, the sum of the areas of the smaller and larger pastures is:
x * y + 4xy = 2700
Simplifying the equation gives:
5xy = 2700
Step 6: Now, we want to find the dimensions that minimize the amount of fencing required. The amount of fencing required is equal to the perimeter of both pastures combined. The perimeter of a rectangle is given by the formula:
perimeter = 2(length + width)
For the smaller pasture, the perimeter would be:
2(x + y)
For the larger pasture, the perimeter would be:
2(2x + 2y) = 4(x + y)
So, the total perimeter would be:
2(x + y) + 4(x + y) = 6(x + y)
Step 7: To find the dimensions that minimize the amount of fencing, we need to minimize the total perimeter while satisfying the area constraint. We can substitute the area equation we derived earlier into the total perimeter equation:
6(x + y) = 6 * (2700 / 5)
Simplifying the equation gives:
6(x + y) = 3240
Dividing both sides by 6:
x + y = 540
Step 8: Since the question asks for the dimensions of the larger pasture, which is 2x and 2y, we can substitute the value of x + y we obtained in the previous step:
2x + 2y = 2 * 540
Simplifying the equation gives:
2x + 2y = 1080
Therefore, the dimensions of the larger pasture should be 2x km by 2y km, which is equal to 1080 km by 1080 km.