niran has 72 centimeters of molding to make a frame for a print. this is not enough molding to frame the entire print.How shouldhe cut the molding to give the largest possible area for the print using the inside edge of the molding as the perimeter?

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niran has 72 centimeters of molding to make a frame for a print. this is not enough molding to frame the entire print.How shouldhe cut the molding to give the largest possible area for the print using the inside edge of the molding as the perimeter?

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To determine how Niran should cut the molding to give the largest possible area for the print, let's break down the problem step by step.

1. Understand the problem:
Niran has 72 centimeters of molding, and he wants to use it to create a frame. However, the molding is not enough to frame the entire print. Niran wants to cut the molding to give the largest possible area for the print using the inside edge of the molding as the perimeter.

2. Determine the shape of the frame:
To maximize the area for the print, Niran needs to cut the molding into a rectangular shape. This is because rectangles have the highest area for a given perimeter compared to other shapes like squares or circles.

3. Define the variables:
Let's assume the length of the molding will be x centimeters, which will be used as the width of the rectangular frame. The remaining molding (72 - x) centimeters will be used as the length of the frame.

4. Write an equation:
The equation to calculate the area of a rectangle is: Area = Length × Width.

5. Maximize the area:
To find the largest possible area, we need to find the maximum value of the area equation. Since we have defined the width (x), we can express the length of the frame as (72 - x).

Area = x × (72 - x)

6. Find the maximum area:
We can find the maximum area by examining the graph of this equation or using calculus. To keep it simple, let's use calculus:
- Take the derivative of the area equation with respect to x: d/dx (x × (72 - x))
dA/dx = 72 - 2x
- Set the derivative equal to zero to find the critical points:
72 - 2x = 0
2x = 72
x = 36

Therefore, the critical point is x = 36 centimeters.

7. Check the endpoints and critical points:
To determine which value gives the maximum area, we need to evaluate the area equation at the endpoints (0 and 72) as well as the critical point (36):
- Endpoint 0:
Area = 0 × (72 - 0) = 0
- Endpoint 72:
Area = 72 × (72 - 72) = 0
- Critical point 36:
Area = 36 × (72 - 36) = 36 × 36 = 1296

The largest possible area for the print is 1296 square centimeters, achieved when Niran cuts the molding into a rectangular frame with a width of 36 centimeters and a length of (72 - 36) = 36 centimeters.