Function Problem: Area of Rectangle with 100ft Perimeter

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In summary, the conversation discusses expressing the area of a rectangle with a perimeter of 100 feet as a function of the length of one of its sides. The function for area can be written as A = w(50-w) or A = -w^2 + 50w, with width as the subject of the perimeter equation.
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Arreat
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Homework Statement



Express the area of a rectangle with the perimeter of 100 feet as a function of the lengh L of one of its sides


Homework Equations


100=2w+2l


The Attempt at a Solution


As far as I can get is,

L = 50 -2w

Just writing the function is a problem, I assume it's something like but I'm not sure, sorry if I'm completely off it's been a long day.
A(W) = (50-2w)(w) = 50w - 2w^2
or
A = w(50-w)
 
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  • #2
You successfully wrote the function for area (in two equivalent forms).
 
  • #3
If [tex]100=2L+2W[/tex]

then [tex]L=50-W[/tex] but [tex]L\neq 50-2W[/tex]

Therefore your 2nd equation for the function is the correct one. [tex]A=-W^2+50W[/tex]

But the question asked as a function of the length, not width. It will be similar though. Just make width the subject of the perimeter equation and substitute into the area formula.
 
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Related to Function Problem: Area of Rectangle with 100ft Perimeter

1. What is the formula for finding the area of a rectangle with a perimeter of 100 feet?

The formula for finding the area of a rectangle is length x width. Since we know the perimeter is 100 feet, we can use the formula 2(length + width) = 100 to find the length or width. Once we have the length or width, we can plug it into the area formula to find the total area of the rectangle.

2. How do I solve for the length and width of a rectangle with a perimeter of 100 feet?

To solve for the length and width, we can use the formula 2(length + width) = 100. We can also rearrange the formula to solve for either the length or width, depending on which one we are missing. Once we have either the length or width, we can plug it into the area formula to find the total area of the rectangle.

3. Can I use any measurements for the length and width of the rectangle as long as the perimeter is 100 feet?

Yes, you can use any measurements for the length and width of the rectangle as long as they add up to a perimeter of 100 feet. For example, you could have a rectangle with sides of 25 feet and 50 feet, or sides of 30 feet and 40 feet. As long as the length and width add up to 100 feet, the area will be the same.

4. What is the maximum area that can be achieved with a perimeter of 100 feet?

The maximum area that can be achieved with a perimeter of 100 feet is when the rectangle is a square. This means that the length and width are equal, and each side would be 25 feet. The formula for finding the area of a square is length x length, or length squared. Therefore, the maximum area would be 25 feet x 25 feet = 625 square feet.

5. Can I use this formula to find the area of any rectangle with a perimeter of 100 feet?

Yes, you can use this formula to find the area of any rectangle with a perimeter of 100 feet. As long as you know the perimeter, you can use the formula 2(length + width) = 100 to find the length or width. Once you have either the length or width, you can plug it into the area formula to find the total area of the rectangle.

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