arnold has 24m of fencing to surround a garden, bounded on one side by the wall of his house. what are the dimensions of the largest rectangular garden that he can enclose?
how do i solve this question
please help
step by step solutions please
A square is the largest area for a given perimeter, if all sides are included. In this case, one side is not, so suppose the side parallel to the wall has length x, and the other sides have length y.
x+2y = 24
The area is
a = xy = (24-2y)y = 24y-2y^2
You know that this is just a parabola, with its vertex where y = 24/4 = 6.
So, y=6 and x=12, with area=72
An 8x8 square would only have area 64.
why did u multiply the whole thing by "y" for (24-2y)y ?
Well, first of all, I have to say that Arnold must be quite the green thumb to be able to garden on his house wall! But let's focus on solving this problem step by step.
To find the dimensions of the largest rectangular garden, we need to maximize the area while using up all 24 meters of fencing.
Let's call the length of the garden L and the width W.
Step 1: Create an equation using the given information.
We know that the perimeter of a rectangle is given by the formula P = 2L + 2W. In this case, the perimeter is 24 meters. Since the garden is bounded on one side by the wall of his house (which is already accounted for), we can simplify the equation to P = L + 2W.
Given: P = 24m
Step 2: Rewrite the equation in terms of only one variable.
Since we want to find the maximum area, we need to express the equation in terms of only one variable. Let's solve the equation for L:
L = 24 - 2W
Step 3: Substitute the value of L in terms of W into the area equation.
The area of a rectangle is given by the formula A = L * W. Substitute the value of L we found above into the area equation:
A = (24 - 2W) * W
Step 4: Simplify the equation and find the maximum area.
Expand the equation: A = 24W - 2W^2
Rearrange it: A = -2W^2 + 24W
Now, this is a quadratic equation with a negative leading coefficient, which means it represents a parabola that opens downwards.
To find the dimensions that maximize the area, we need to find the vertex of the parabola. The x-value of the vertex gives us the value of W, while the y-value gives us the maximum area.
Step 5: Find the value of W that maximizes the area.
The x-value of the vertex can be found using the formula x = -b / 2a, where a, b, and c are the coefficients of the quadratic equation ax^2+bx+c = 0.
In this case, a = -2 and b = 24, so we have:
W = -24 / (2*-2)
W = -24 / -4
W = 6
Step 6: Find the maximum area using the value of W.
Now that we found the value of W, substitute it back into the equation for the area:
A = -2(6)^2 + 24(6)
A = -72 + 144
A = 72
So, the maximum area is 72 square meters.
Step 7: Find the dimensions of the garden.
Since we found W = 6, substitute it back into the equation we found for L:
L = 24 - 2(6)
L = 12
Therefore, the dimensions of the largest rectangular garden that Arnold can enclose are L = 12m and W = 6m.
Hope that helps, and happy gardening!
To solve this question step by step, you can follow these guidelines:
1. Understand the problem: In this question, Arnold has 24 meters of fencing to surround a garden. One side of the garden is already bounded by the wall of his house. The task is to find the dimensions of the largest rectangular garden that he can enclose.
2. Formulate the problem mathematically: Let's say the length of the garden is 'L' meters and the width is 'W' meters. The total amount of fencing required is the perimeter, which is given as 24 meters (2 sides of length 'L' and 2 sides of length 'W').
Perimeter = 2L + 2W = 24 meters
3. Simplify the expression: Rearrange the equation to express one variable in terms of the other. Let's solve the equation for 'L':
2L = 24 - 2W
L = (24 - 2W) / 2
L = 12 - W
4. Deduce the equation: The area of a rectangle is calculated by multiplying its length (L) by its width (W). We want to maximize the area, given the constraint of 24 meters of fencing.
Area = L * W
Area = (12 - W) * W
5. Maximize the area: To find the maximum area, we can analyze the graph of the equation. Since the equation is quadratic, the graph will represent a downward-opening parabola. The maximum area will occur at the vertex of the parabola.
6. Find the vertex: The x-coordinate of the vertex of a quadratic equation in the form Ax^2 + Bx + C can be found using the formula: x = -B / (2A). In our case, the equation is in the form -W^2 + 12W.
x = -B / (2A)
x = -(-12) / (2 * -1)
x = 12 / 2
x = 6
So, W = 6 meters.
7. Substitute the value of 'W' into the expression for 'L': From earlier, we had L = 12 - W.
L = 12 - 6
L = 6 meters.
8. Answer: The maximum rectangular garden that Arnold can enclose has dimensions of 6 meters (width) by 6 meters (length).
24/3 is 8
A square is the largest area.
24/3 = ?