A rectangular yard with an area of 50m squared is to be fenced on three sides. Minimizing the perimeter will minimize the cost of the fence. Conduct an investigation to determine the shape 0f the yard with the minium perimeter.
I sqaure rooted the 50m and I got 7.07 which then I added 3 times to get the perimeter since has only three sides which then i got 21.21 and i went completely wrong from after that when i double check the answer.... appearetly the real answer is 5m by 10m and I don't how they got it when i was taught to square root the area when you dont know the perimeter...
determine the shape? to minimize perimeter? And you say 5x10?
The minimum perimeter for any geometric shape is circle, in this case with a side already fenced, the shape will be a semicircle.
radius of semicircle:
50=1/2 PI r^2
r= sqrt (10/pi)
minimum perimeter= PI(radius)
= pi sqrt (10/pi)=sqrt10pi
if it is a circle why does it say RECTANGUlAR yard with a area of 50m sqaured
It started out as a rectangular shape. I have no idea why the problem was worded this way. To me, if they wanted it to remain a rectangular shape, they would have asked for the new dimensions of the rectangle of area 50 with three sides fenced.
So I have no idea why it is worded, but the shape to minimize such an area will be a shape of a semicircle.
Could be the teacher didn't mean "determine the shape"
square rooting implies that the yard is square ... from the result of your answer, this is not the case
the "perimeter" in this problem, is only three sides
A = L * W = 50
P = L + 2W
a good investigation might consist of a table comparing L, W, and P
... remember, you're looking for a minimum P
To determine the shape of the yard with the minimum perimeter, we need to understand the problem and use some mathematical reasoning.
Given that the yard has an area of 50m² and is to be fenced on three sides, we want to find the dimensions of the yard that will minimize the perimeter, which in turn minimizes the cost of the fence.
Let's begin by assuming that the yard is a rectangle with length L and width W.
The area of the rectangle is given by the formula: Area = Length × Width.
In this case, we have Area = 50m².
If we are to fence three sides of the rectangle, that means only the length and two widths will require fencing.
The perimeter of the rectangle is given by the formula: Perimeter = 2×Length + 2×Width.
However, since we are only to fence three sides, we can modify this formula to: Perimeter = Length + 2×Width.
Now, let's substitute the values into our formulas.
From Area = Length × Width, we have 50 = L × W.
From Perimeter = Length + 2×Width, we need to express one variable in terms of the other to have a single variable equation.
Let's solve the area equation for one of the variables:
W = 50 / L.
Substitute this expression into the perimeter formula:
P = L + 2 × (50 / L).
To minimize the perimeter, we can differentiate this equation with respect to L and set the derivative equal to zero.
dP/dL = 1 - 100 / L² = 0.
Solving this equation for L, we find L = 10m.
Substituting this value back into W = 50 / L, we find W = 5m.
Therefore, the dimensions of the yard with the minimum perimeter are 10m by 5m.
To summarize the steps:
1. Start with the area equation: 50 = L × W.
2. Substitute W = 50 / L into the perimeter equation: P = L + 2 × (50 / L).
3. Differentiate the perimeter equation and set the derivative equal to zero.
4. Solve the resulting equation to find L.
5. Substitue L back into W = 50 / L to find W.
I apologize for any confusion caused by the initial incorrect calculation.