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Darina [25.2K]
3 years ago
14

Find the area of the trapezoidal cross-section of the irrigation canal shown below. Your answer will be in terms of h, w, and θ.

Mathematics
2 answers:
Verdich [7]3 years ago
8 0

The area of the trapezium is \boxed{\left({w+\frac{h}{{\tan \theta }}}\right)h} .

Further explanation:

The area is the trapezium can be calculated as,

  {\text{area of trapaezium}}=\frac{1}{2}\times\left({{\text{sum of parallel sides}}}\right)\times{\text{height}}

The trigonometry ratio used in the right angle triangles.

The tangent ratio can be written as,

\tan \theta  = \frac{{{\text{length of the side opposite to }}\theta }}{{{\text{length of the side adjacent to }} \theta }}

Here, base is the length of the side adjacent to angle \theta  and the length of side opposite to angle \theta  is perpendicular.

Step by step explanation:

Step 1:

Do naming the given trapezium as attached in the figure.

After naming we have PQRS  is a trapezium  

Consider h  as the height of the trapezium in which {\text{X and W}}  are the points drawn on the line segment RS .

The measurement of one of the parallel side PQ  is  w

It can be seen from the attached figure that RQ{\text{ and }}SP  are the transversal lines.

Therefore, by alternate interior angle property \angle QRW=\theta{\text{ and }}\angle PSX=\theta .

The another parallel side can be written as SR=SX+XW+WR .

Consider the sides as XW=w,SX=x,WR=x .

Therefore, the parallel side can be written as SR=x+w+x .

Step 2:

Now in the right angle triangle XSP  the tangent ratio can be applied as,

\begin{aligned}\tan \theta &=\frac{{{\text{length of the side opposite to }}\theta }}{{{\text{length of the side adjacent to}}\ \theta }}\hfill\\\tan \theta &=\frac{h}{x}\hfill\\x&=\frac{h}{{\tan \theta }}\hfill\\\end{aligned}

Now substitute the value of x  in the equation SR=x+w+x .

\begin{gathered}SR=\frac{h}{{\tan \theta }}+w+\frac{h}{{\tan \theta }}\hfill\\SR=2\frac{h}{{\tan \theta }}+w\hfill\\\end{gathered}

Step 3:

The area of the trapezium can be calculated as,

\begin{aligned}{\text{area of trapaezium}}&=\frac{1}{2}\times\left({{\text{sum of parallel sides}}}\right)\times{\text{height}}\\&=\frac{1}{2}\times\left({PQ+SR}\right)\times h\\&=\frac{1}{2}\times\left({w+w+\frac{{2h}}{{\tan \theta}}}\right)\times h\\&=\frac{1}{2}\times\left({2w+\frac{{2h}}{{\tan \theta }}}\right)\times h\\\end{aligned}

Further simplify the above equation.

\begin{gathered}{\text{area of trapaezium}}=\frac{1}{2}\times\left({2w+\frac{{2h}}{{\tan\theta}}}\right)\times h\hfill\\{\text{area of trapaezium}}=\left({w+\frac{h}{{\tan \theta }}}\right)h\hfill \\\end{gathered}

Therefore, the area of the trapezium is \left({w+\frac{h}{{\tan\theta }}}\right)h .

Learn more:  

  • Learn more about the distance between two points on the number line <u>brainly.com/question/6278187 </u>
  • Learn more about the distance between two coordinates of the line <u>brainly.com/question/10135690 </u>
  • Learn more about what is the domain of the function on the graph? all real numbers all real numbers greater than or equal to 0 all real numbers greater than or equal to –2 all real numbers greater than or equal to –3 <u>brainly.com/question/3845381 </u>

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Perimeters and area

Keywords: trapezium, area, parallel sides, sum, height, length, opposite side, adjacent side, trigonometry, tangent ratio, perpendicular, base, hypotenuse, right angle triangle.

miskamm [114]3 years ago
3 0

[w + (h/tan θ)] × h

<h3>Further explanation</h3>

We aim is to find the area of the trapezoidal cross-section of the irrigation canal.

The formula for the area of the trapezoid is \boxed{\boxed{ \ Area = \frac{1}{2} \times (a + b) \times h \ }}

  • a & b = parallel sides
  • h = height

We assume the lower side is \boxed{ \ a = w \ } and the upper side is b = x + w + x or \boxed{ \ b = w + 2x \ }. See the attached picture.

In the right triangles located on the left and right of the trapezoid, we calculate the value of x based on trigonometric ratios namely tan of theta.

\boxed{ \ tan \ \theta = \frac{opposite}{adjacent} \ }

\boxed{ \ tan \ \theta = \frac{h}{x} \ }

\boxed{ \ x = \frac{h}{tan \ \theta} \ }

Substitute the equation of x to \boxed{ \ b = w + 2x \ }.

We get \boxed{ \ b = w + \frac{2h}{tan \ \theta} \ }

Finally all components are complete and can be substituted into a formula to calculate the area of a trapezoid.

\circ \ \boxed{a = w}\\ \circ \ \boxed{ \ b = w + \frac{2h}{tan \ \theta} \ }\\\circ \ and \ h

\boxed{ \ Area = \frac{1}{2} \times (w + w + \frac{2h}{tan \ \theta}) \times h \ }

\boxed{ \ Area = \frac{1}{2} \times (2w + \frac{2h}{tan \ \theta}) \times h \ }

\boxed{ \ Area = \frac{1}{2} \times 2(w + \frac{h}{tan \ \theta}) \times h \ }

Hence, the area of the trapezoidal cross-section of the irrigation canal is \boxed{\boxed{ \ Area = (w + \frac{h}{tan \ \theta}) \times h \ }}

<h3>Learn more</h3>
  1. Find out the area of parallelogram brainly.com/question/4459688
  2. The order of rotational symmetry of rhombus  brainly.com/question/4228574
  3. Find out the coordinates of the image of a point after the triangle is rotated 270° about the origin brainly.com/question/7437053

Keywords: the trapezoidal, cross-section of the irrigation canal, the area, the answer will be in terms of h, w, and θ, trigonometric ratios, tan, opposite, adjacent, height, sides

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