Tessa has 80 ft of fencing available to construct a fence that will divide her garden into four equal rectangular sections. Her house forms one side of the garden and 𝑥 represents the width, as shown in Figure 1.
(a) Express the total area of the four sections as a function of x.
Hint: if you call 𝑦 each of the four sections perpendicular to 𝑥, find the perimeter of the fence in terms of 𝑥,𝑦 x,y, then use the fact that the perimeter is 80 ft.
𝐴(𝑥)=
Don't know what the diagram looks like but I assumed
a rectangle against the wall, (one length not needed)
I then split that rectangle into 4 equal rectangles
For a small rectangle , let the width be x and its length by y.
So I will need 6x + 4y = 80
3x + 2y = 40
y = (40-3x)/2
area of whole big one = (2x)(2y) = 4xy
= 4x(40-3x)/2
= 80x - 6x^2
d(area) = 80 - 12x = 0 for a max of area
x = 80/12 = 20/3
then y = (40-3x)/2
= (40 - 20)/2 = 10
state you conclusion
(Oh boy, a math question! Let me put on my thinking cap... or should I say, my floppy shoes!)
Alright, let's tackle this one step at a time. We need to calculate the total area of the four sections, so let's start by finding the area of each section.
Since the width of each section is represented by 𝑥, we can say that the height of each section is also 𝑥, as the sections are rectangular. Therefore, the area of each section is 𝑥 * 𝑥 = 𝑥².
Since there are four sections, the total area of all the sections together is 4 times the area of each section. So, the expression for the total area 𝐴(𝑥) would be:
𝐴(𝑥) = 4 * 𝑥²
Voila! The total area of the four sections is 4 times 𝑥 squared. Now, go forth and calculate, my friend, but don't forget to bring your clown nose! 🤡
To express the total area of the four sections as a function of x, we need to find the dimensions of each section in terms of x.
Let's assume the length of each section perpendicular to x is y.
Each section will have two sides of length x and two sides of length y.
The perimeter of each section can be calculated as:
Perimeter = 2x + 2y
Since we have four sections, the total perimeter of the fence will be 4 times the individual section perimeter:
Total Perimeter = 4(2x + 2y)
= 8x + 8y
Given that the total perimeter is 80 ft, we can set up the equation:
8x + 8y = 80
To find y in terms of x, we can solve this equation for y:
8y = 80 - 8x
y = (80 - 8x) / 8
y = 10 - x
Now, the area of each section can be calculated as the product of its length and width:
Area = x * y
Area = x * (10 - x)
Since we have four sections, the total area of the four sections can be expressed as a function of x:
A(x) = 4 * (x * (10 - x))
A(x) = 4x(10 - x)
Thus, the total area of the four sections as a function of x is A(x) = 4x(10 - x).
To express the total area of the four sections as a function of x, we need to find the dimensions of each section.
Let's break down the problem:
1. We have a garden divided into four equal rectangular sections.
2. The house forms one side of the garden.
3. The width of each section is represented by x.
4. The height of each section (perpendicular to x) is represented by y.
To find the perimeter of the fence, we need to consider all the sides of the garden:
- The length of the garden is (x + 2y) because there are two y-height sections on each side of x.
- The width of the garden is (x + y) because there is one y-height section on top of x.
- The perimeter of the garden can be calculated as: 2(x + 2y) + 2(x + y) = 4x + 4y + 2x + 2y = 6x + 6y.
We are given that the total length of the fencing available is 80 ft. Equating this to the perimeter of the garden, we have:
6x + 6y = 80
Now, we need to express the area of the four sections, A(x), as a function of x.
The area of each section is simply the width multiplied by the height. So, the area of one section is xy.
Since there are four sections, the total area A(x) will be 4xy.
To summarize:
𝐴(𝑥) = 4xy
To find the specific value of 𝐴(𝑥), you will need to solve the equation 6x + 6y = 80 for y in terms of x. Then substitute that value into the expression 4xy to get 𝐴(𝑥) in terms of x.