# Maximizing area of trapezoid Drain Gutter

• testme
In summary, the homework statement asks for the area of a trapezoid with a given length and width, given that the bottom and sides of the trapezoid must add up to 60 cm. The student is not sure how to solve the problem, and asks for help.
testme

## Homework Statement

Image is very big.

Angles A and B are 120°
length of metal is 5m
width of metal is 0.6m
it's an isosceles trapezoid drainage.

## Homework Equations

° in a trapezoid: 180(n-2)
A = 2(bh/2) + wh

## The Attempt at a Solution

I'm not sure, I started but I'm not even sure if I'm thinking of it right since the question is stated in a confusing way. This is what I believe I have to do so far.

w = AD + AB + BC = 0.6
l = 2x + 2y = 5
x = z + 2b

A = bh + wh

I know it's really complicated but I'm not quite sure how to go about it, I'm just wondering if someone can tell me exactly what the question is asking and if I'm on the right track, thanks.

testme said:

## Homework Statement

Image is very big.

Angles A and B are 120°
length of metal is 5m
width of metal is 0.6m
it's an isosceles trapezoid drainage.

## Homework Equations

° in a trapezoid: 180(n-2)
A = 2(bh/2) + wh

## The Attempt at a Solution

I'm not sure, I started but I'm not even sure if I'm thinking of it right since the question is stated in a confusing way. This is what I believe I have to do so far.

w = AD + AB + BC = 0.6
l = 2x + 2y = 5
x = z + 2b

A = bh + wh

I know it's really complicated but I'm not quite sure how to go about it, I'm just wondering if someone can tell me exactly what the question is asking and if I'm on the right track, thanks.

It is impossible to tell if you are on the right track, because you do not define your symbols. For example, in the equation A = bh + wh, is A = area, or is A = angle A, or what? What are b,h,w? In the equation 2x + 2y = 5, what are x and y supposed to represent?

In this problem it is clear that the length 5m is irrelevant; you are asked to maximize the cross-sectional area, subject to bottom + sides adding up to 60 cm.

RGV

Sorry about that.

A = bh + wh
where b is the b of the triangles if you cut up the trapezoid and h is the height. w is the width of the top after the two triangle bases have been taken out.

So then it would have to be that

top + bottom + 2side = 0.6 m?
the top = bottom + 2base

so then could it be written as

0.6 = 2bottom + 2side + 2base?

Let's assume that AB is the bottom of the trapezoid. Then there is no top! The gutter doesn't have a top to it. So just bottom+2*side = .6

Aso, it's not true that top=bottom+2*base, you need to use a little trigonometry to figure out what the relationship really is

In this problem the only real control that you have is how long AB is - this determines every other aspect of the gutter. So your objective is to write the area function purely as a function of AB by using the constraints in the problem... in this case we have

area=wb+hb

w is just the length of AB, so you need to describe the values of b and h in terms of the length of AB

testme said:
Sorry about that.

A = bh + wh
where b is the b of the triangles if you cut up the trapezoid and h is the height. w is the width of the top after the two triangle bases have been taken out.

So then it would have to be that

top + bottom + 2side = 0.6 m?
the top = bottom + 2base

so then could it be written as

0.6 = 2bottom + 2side + 2base?

No: look at the figure; the top is open, so no metal is used on the top. You still need to figure out the area.

RGV

Using w as the length of AB, h as the height made, and b would then be the base. would it be

Area = wh + 2(bh/2)
Area = wh + bh

Then for 0.6 being the width of the entire sheet

0.6 = BC + AB + AD
BC = AD, let x = BC = AD
0.6 = AB + 2x
we'll use w again for AB
0.6 = w + 2x

I got the right answer, it's not too hard once I understood exactly what they meant and a little bit of a push in the right direction. Though working with sines of angles isn't my favorite. Thanks again :)

## 1. How do you calculate the area of a trapezoid drain gutter?

In order to calculate the area of a trapezoid drain gutter, you will need to know the length of both parallel sides (a and b) and the height of the trapezoid (h). The formula for calculating the area is: (a + b) / 2 * h. Make sure to use consistent units of measurement for accurate results.

## 2. What is the maximum area of a trapezoid drain gutter?

The maximum area of a trapezoid drain gutter depends on the dimensions of the trapezoid. However, the general rule is that the longer the parallel sides and the greater the height, the larger the area will be. There is no specific maximum area as it can vary depending on the specific dimensions.

## 3. How can you increase the area of a trapezoid drain gutter?

The area of a trapezoid drain gutter can be increased by increasing the length of both parallel sides or by increasing the height of the trapezoid. However, it is important to consider the limitations of the space where the gutter will be installed and to ensure that the gutter is still functional for its intended purpose.

## 4. Are there any other factors to consider when maximizing the area of a trapezoid drain gutter?

Yes, there are other factors to consider when maximizing the area of a trapezoid drain gutter. These include the material and thickness of the gutter, the slope of the gutter, and the placement of the gutter in relation to the surrounding landscape or building structure. It is important to ensure that the gutter is structurally sound and able to efficiently drain water.

## 5. Can the area of a trapezoid drain gutter be maximized without compromising its functionality?

It is possible to maximize the area of a trapezoid drain gutter without compromising its functionality, but it requires careful planning and consideration of all factors involved. It is important to consult with a professional or conduct thorough research to determine the best dimensions and placement for the drain gutter in order to maximize its area while still maintaining its effectiveness.

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