suppose you are enclosing a rectangular area to create a rabbit cage. you have 80 ft of fence and want to build a pen with the largest possible area for your rabbit, so you build the cage using an existing building as one side.
make a table showing the areas for some selected values of x, and write a function that gives the area, y, as a function of the width, x
Let x be the width of the side parallel to the building. The other two sides then have length (80-x)/2
The area is y = x(40 - x/2)
y = 40x - (x^2/2)
You get maximum area when dy/dx = 0
40 -x = 0
x = 40
Table:
x y
20 600
30 750
39 799.5
40 800
41 799.5
50 750
60 600
afh
To find the largest possible area for the rabbit cage, we need to consider that one side of the cage will be the existing building, while the other three sides will be made up of the fence. Let's assume the width of the cage (perpendicular to the building) is x and the length (parallel to the building) is y.
To start, we know that the total amount of fence we have is 80 ft. Therefore, the sum of the lengths of all three sides, excluding the building side, should be 80 ft. Mathematically, this can be represented as:
2x + y = 80 --> equation 1
To find the area of the rectangular cage, we multiply the width (x) by the length (y). So the area, A, can be expressed as:
A = xy
To create a table showing the areas for selected values of x and determine the function for the area, let's consider some values for x and find the corresponding y and A:
Table:
|x | y = 80 - 2x | A = xy |
|----|-------------|----------------|
|5 | 70 | 5 * 70 = 350 |
|10 | 60 | 10 * 60 = 600 |
|15 | 50 | 15 * 50 = 750 |
|20 | 40 | 20 * 40 = 800 |
|25 | 30 | 25 * 30 = 750 |
|30 | 20 | 30 * 20 = 600 |
|35 | 10 | 35 * 10 = 350 |
From the table, we can observe that as x increases, y decreases. The area (A) reaches its maximum when x is 20, having a value of 800 square feet.
We can now write the function that gives the area, y, as a function of the width, x:
y = 80 - 2x
Therefore, the area of the rabbit cage in terms of the width (x) is given by:
A(x) = x * (80 - 2x)
This function will give you the maximum possible area for the rabbit cage.
To solve this problem, we need to understand that the area of the rectangular pen is the product of its length and width. Let's denote the length of the pen as y and the width as x.
Now, let's construct a table by substituting different values of x (the width) into the area formula and calculate the corresponding areas (y):
| Width (x) | Area (y) |
|-----------|----------|
| 20 ft | 1600 ft² |
| 30 ft | 2400 ft² |
| 40 ft | 3200 ft² |
| 50 ft | 4000 ft² |
| 60 ft | 4800 ft² |
| 70 ft | 5600 ft² |
| 80 ft | 6400 ft² |
To find a function that expresses the area (y) as a function of the width (x), we can observe that the area is directly proportional to the width. So, we can write the formula as follows:
y = k * x,
where k is a constant factor.
To find the value of k, we can use any of the given data points. Let's use the point (x = 40 ft, y = 3200 ft²):
3200 = k * 40,
Dividing both sides by 40:
k = 3200 / 40 = 80.
Therefore, the function that gives the area (y) as a function of the width (x) is:
y = 80 * x.