A rectangular lot is bordered on one side by a stream and on the other 3 sides by 600m of fencing. The area is a maximum. Determine the area.
Let the width be x
let the length by y
2x + y = 600
y = 600-2x
Area = xy
= x(600-2x)
= 600x - 2x^2
If you know Calculus ...
d(area)/dx = 600 - 4x
= 0 for a max/min
4x = 600
x = 150
so Area = 150(600-300) = 45000 m^2
If you don't know calculus ...
complete the square ..
Area
= -2(x^2 - 300x + 22500 - 22500)
= -2((x - 150)^2 - 22500)
= -2(x-150)^2 + 45000
can you read the information from this last line ?
Why did the fence become an expert mathematician? Because it's always trying to find the maximum area!
For this problem, let's use our math skills to maximize the area of the rectangular lot.
Given that the lot is bordered on one side by a stream and on the other three sides by 600m of fencing, we can visualize it as a rectangle with one side on the stream and the other three sides fenced.
Let's denote the length of the side along the stream as x, and the two adjacent sides as y. The fourth side will also be y, forming a rectangle.
Since the total length of the three fenced sides is 600m, we have the equation:
2y + x = 600
We want to maximize the area of the rectangle, which is given by:
A = x * y
To find the area in terms of a single variable, we can rewrite the equation for x as:
x = 600 - 2y
Substituting this into the equation for area, we get:
A = (600 - 2y) * y
Now, let's multiply out the terms:
A = 600y - 2y^2
To maximize the area, we can take the derivative of A with respect to y and set it equal to zero:
dA/dy = 600 - 4y = 0
Solving this equation, we find:
600 - 4y = 0
4y = 600
y = 150
Now, substitute this value back into the equation for x:
x = 600 - 2y
x = 600 - 2(150)
x = 600 - 300
x = 300
So, the dimensions of the rectangular lot that maximize the area are x = 300m and y = 150m.
Finally, we can calculate the area using these values:
A = x * y
A = 300 * 150
A = 45,000 square meters
Therefore, the maximum area of the rectangular lot is 45,000 square meters.
To determine the maximum area of the rectangular lot, we need to consider the relationship between the length and width of the lot. Let's break it down step-by-step:
Step 1: Assign variables
Let's assume the length of the lot is L and the width is W.
Step 2: Write an equation for the perimeter
Since the lot is bordered on one side by a stream and on the other three sides by 600m of fencing, we can write an equation for the perimeter of the lot:
2L + W = 600 (Equation 1)
Step 3: Write an equation for the area
The area of the rectangular lot can be calculated by multiplying the length (L) by the width (W):
Area = L * W (Equation 2)
Step 4: Solve Equation 1 for L
Rearranging Equation 1, we can solve for L:
L = (600 - W) / 2
Step 5: Substitute the value of L into Equation 2
Now, substitute the value of L (from step 4) into Equation 2:
Area = (600 - W) / 2 * W
Step 6: Simplify the area equation
Distribute the division by 2:
Area = (600W - W^2) / 2
Step 7: Find the maximum area
To find the maximum area, we need to find the vertex of the quadratic equation. The equation is in the form of Ax^2 + Bx + C. In our equation, A = -1, B = 600, and C = 0.
The x-coordinate of the vertex can be found using the formula: x = -B/2A.
In our case, x = -600 / 2(-1) = 300
Step 8: Substitute the value of x into the area equation
Plug the value of x back into the equation for the area:
Area = (600(300) - (300)^2) / 2
= (180000 - 90000) / 2
= 90000 m^2
Therefore, the maximum area of the rectangular lot is 90000 square meters.
To determine the maximum area of the rectangular lot, we can use the concept of optimization. Let's break down the problem and come up with a solution.
1. Understand the problem:
- We have a rectangular lot with one side bordered by a stream.
- The lot is fenced on the other three sides with a total of 600m of fencing.
- We need to find the maximum possible area of the lot.
2. Identify the variables:
- Let's assume the length of the lot parallel to the stream is "x" meters.
- The width, perpendicular to the stream, can be represented as "y" meters.
3. Formulate the equation:
- The equation for the perimeter of the lot can be written as: P = 2x + y = 600m
- To isolate y, we rearrange the equation: y = 600m - 2x
4. Write the formula for the area:
- The area of the lot can be calculated by multiplying the length and width: A = x * y
5. Express the area equation in terms of a single variable:
- Substituting the value of y from earlier, the area equation becomes: A = x * (600m - 2x)
- Simplifying further: A = 600x - 2x^2
6. Find the vertex of the area equation:
- The maximum area occurs at the vertex of the graph.
- The x-coordinate of the vertex can be found using the formula: x = -b / (2a)
- Here, a = -2 and b = 600.
- Plugging in the values, we get: x = -600 / (2 * -2) = 150m
7. Calculate the maximum area:
- Substitute the x-value into the area equation: A = 600 * 150 - 2 * 150^2
- Simplifying: A = 90000m^2
So, the maximum possible area of the rectangular lot is 90000 square meters.