Math / Geometry

QuestionHeather has 360 meters of fencing and wishes to enclose a rectangular field. Suppose that a side length (in meters) of the field is $x$ , as shown below. (a) Find a function that gives the area $A(x)$ of the field (in square meters) in terms of $x$ . $A(x)=$ $\square$ (b) What side length $x$ gives the maximum area that the field can have?
Side length $x$ : $\square$ meters (c) What is the maximum area that the field can have?
Maximum area: $\square$ square meters

Studdy Solution

STEP 1

What is this asking? We need to figure out the biggest rectangular area Heather can make with her fencing. Watch out! Don't mix up the length of the fence with the area of the field!

STEP 2

1. Express the other side in terms of $x$ .
2. Define the area function.
3. Optimize the area function.
4. Calculate the maximum area.

STEP 3

We know Heather has 360 meters of fencing.
A rectangle has two sides of length $x$ , so the length of the two remaining sides combined is $360 - 2 \cdot x$ .
Since the remaining two sides are equal, each side has length $(360 - 2 \cdot x) / 2 = 180 - x$ .

STEP 4

The area of a rectangle is length times width.
We just figured out that one side is $x$ and the other is $180 - x$ .
So, the area $A(x)$ is $x \cdot (180 - x)$ .

STEP 5

Let's expand this: $A(x) = 180 \cdot x - x^2$ .
There we go!
Our area function is ready.

STEP 6

To find the maximum area, we need to find the vertex of this parabola.
The x-coordinate of the vertex of a parabola $ax^2 + bx + c$ is given by $-b / (2 \cdot a)$ .

STEP 7

In our area function, $A(x) = -x^2 + 180 \cdot x$ , we have $a = -1$ and $b = 180$ .
So, the x-coordinate of the vertex is $-180 / (2 \cdot (-1)) = 90$ .
This means a side length of $x = \mathbf{90}$ meters will give us the maximum area.

STEP 8

Now, we just plug our optimal $x$ value back into our area function: $A(90) = 180 \cdot 90 - 90^2 = 16200 - 8100 = 8100$ .

STEP 9

(a) $A(x) = 180 \cdot x - x^2$ (b) Side length $x$ : 90 meters (c) Maximum area: 8100 square meters

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Studdy Solution

STEP 1

What is this asking? We need to figure out the biggest rectangular area Heather can make with her fencing. Watch out! Don't mix up the length of the fence with the area of the field!

STEP 2

1. Express the other side in terms of $x$ .
2. Define the area function.
3. Optimize the area function.
4. Calculate the maximum area.

STEP 3

We know Heather has 360 meters of fencing.
A rectangle has two sides of length $x$ , so the length of the two remaining sides combined is $360 - 2 \cdot x$ .
Since the remaining two sides are equal, each side has length $(360 - 2 \cdot x) / 2 = 180 - x$ .

STEP 4

The area of a rectangle is length times width.
We just figured out that one side is $x$ and the other is $180 - x$ .
So, the area $A(x)$ is $x \cdot (180 - x)$ .

STEP 5

Let's expand this: $A(x) = 180 \cdot x - x^2$ .
There we go!
Our area function is ready.

STEP 6

To find the maximum area, we need to find the vertex of this parabola.
The x-coordinate of the vertex of a parabola $ax^2 + bx + c$ is given by $-b / (2 \cdot a)$ .

STEP 7

In our area function, $A(x) = -x^2 + 180 \cdot x$ , we have $a = -1$ and $b = 180$ .
So, the x-coordinate of the vertex is $-180 / (2 \cdot (-1)) = 90$ .
This means a side length of $x = \mathbf{90}$ meters will give us the maximum area.

STEP 8

Now, we just plug our optimal $x$ value back into our area function: $A(90) = 180 \cdot 90 - 90^2 = 16200 - 8100 = 8100$ .

STEP 9

(a) $A(x) = 180 \cdot x - x^2$ (b) Side length $x$ : 90 meters (c) Maximum area: 8100 square meters

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Studdy Solution

STEP 1

What is this asking? We need to figure out the biggest rectangular area Heather can make with her fencing. Watch out! Don't mix up the length of the fence with the area of the field!

STEP 2

1. Express the other side in terms of xxx.2. Define the area function.3. Optimize the area function.4. Calculate the maximum area.

STEP 3

STEP 4

The area of a rectangle is length times width.We just figured out that one side is xxx and the other is 180−x180 - x180−x.So, the area A(x)A(x)A(x) is x⋅(180−x)x \cdot (180 - x)x⋅(180−x).

STEP 5

Let's expand this: A(x)=180⋅x−x2A(x) = 180 \cdot x - x^2A(x)=180⋅x−x2.There we go!Our **area function** is ready.

STEP 6

To find the maximum area, we need to find the **vertex** of this parabola.The x-coordinate of the vertex of a parabola ax2+bx+cax^2 + bx + cax2+bx+c is given by −b/(2⋅a)-b / (2 \cdot a)−b/(2⋅a).

STEP 7

STEP 8

Now, we just plug our **optimal** xxx value back into our area function: A(90)=180⋅90−902=16200−8100=8100A(90) = 180 \cdot 90 - 90^2 = 16200 - 8100 = 8100A(90)=180⋅90−902=16200−8100=8100.

STEP 9

(a) A(x)=180⋅x−x2A(x) = 180 \cdot x - x^2A(x)=180⋅x−x2 (b) Side length xxx: 90 meters (c) Maximum area: 8100 square meters

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1. Express the other side in terms of $x$ .
2. Define the area function.
3. Optimize the area function.
4. Calculate the maximum area.

The area of a rectangle is length times width.
We just figured out that one side is $x$ and the other is $180 - x$ .
So, the area $A(x)$ is $x \cdot (180 - x)$ .

Let's expand this: $A(x) = 180 \cdot x - x^2$ .
There we go!
Our area function is ready.

To find the maximum area, we need to find the vertex of this parabola.
The x-coordinate of the vertex of a parabola $ax^2 + bx + c$ is given by $-b / (2 \cdot a)$ .

Now, we just plug our optimal $x$ value back into our area function: $A(90) = 180 \cdot 90 - 90^2 = 16200 - 8100 = 8100$ .

(a) $A(x) = 180 \cdot x - x^2$ (b) Side length $x$ : 90 meters (c) Maximum area: 8100 square meters