a long strip of sheet metal 12 inches wide is to be formed into an open gutter with a rectangular cross-section as shown. what is the maximum possible area of the cross-section?
Bend it down the middle and form into a trough with angle Ø
The area of the cross-section of the trough will be
(1/2)(12)(12) sinØ
= 72 sinØ
d(area)/dØ = 72 cosØ
= 0 for a max of area
72 cosØ = 0
cosØ = 0
Ø = 90° or 270° , the last can be rejected
So when Ø = 90°
the area of the cross section will be
(1/2)(12)(12)sin90 = 72 square inches
To find the maximum possible area of the cross-section of the open gutter, we need to determine the dimensions that will maximize the area.
Let's break down the problem step by step:
1. Identify the shape: The description states that the cross-section of the open gutter is rectangular. This means we are looking for a rectangle that can be formed using the strip of sheet metal.
2. Consider the dimensions: Since the width of the sheet metal is given as 12 inches, let's assume that the rectangular cross-section will have a width of 12 inches.
3. Determine the height: We need to find the height that will result in the maximum possible area. Since the gutter is open, we can assume that the height can stretch infinitely. However, keep in mind that we are limited by the length of the strip of sheet metal.
4. Apply the area formula: The area of a rectangle is given by the formula A = length × width. In this case, the width is given as 12 inches, but we need to find the length. The length will be the height of the rectangle.
5. Identify the constraint: The length of the strip of sheet metal is not provided in the question. Without the length, we cannot calculate the exact maximum area. However, we can still determine the relationship between the length and the area.
6. Visualize the rectangle: Imagine a rectangle with a width of 12 inches and a height of "h" inches. The length of the rectangle will be the same as the height, "h".
7. Find the area function: Given the dimensions, the area function of the cross-section is A = h × 12.
8. Maximize the area: The area will be maximized when the height, "h", is maximized. Since there are no constraints provided in the question, we can assume that the height can be infinitely large. Therefore, the maximum possible area will be infinite.
In conclusion, the maximum possible area of the cross-section is infinite, given that there are no constraints on the length of the strip of sheet metal.