Question

a field is in the shape of trapezium having parallel sides 90 M and 30 M . these sides meet the third side at the right angles .the length of the fourth side is hundred metre. if it costs rupees 4 to plough 1 meter squares of the field .find the total cost of ploughing
the field. (use heron's formula)

Answer 1

Answer :-

we have,

→ DC = AE = 30 m

so,

→ EB = AB - AE = 90 - 30 = 60m .

now, in right angled ∆CEB,

→ EB = 60m

→ CB = 100 m (given)

then,

→ EC = √(100² - 60²) { By pythagoras theorem }

→ EC = √(10000 - 3600)

→ EC = √6400

→ EC = 80 m

therefore,

→ Area of trapezium = (1/2) * sum of parallel side * height = (1/2) * (30 + 90) * 80 = (1/2) * 120 * 80 = 60 * 80 = 4800 m².

hence,

→ 1 m² cost = Rs.4

→ 4800 m² cost = 4800 * 4 = Rs.19200 (Ans.)

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Answer 2

Given:

A field is in the shape of a trapezium having parallel sides 90 m and 30 m. these sides meet the third side at the right angles. the length of the fourth side is a hundred metre. if it costs rupees 4 to plough 1-meter squares of the field.

To find:

The total cost of ploughing the field. (use heron's formula)

Solution:

Construction:- Draw a ⊥ line from D on BC at E.

Since,

side AD // side BE

side AB // side DE

∠A = ∠B = ∠DEB = 90°

∴ ABED is a rectangle

⇒ AB = DE and AD = BE = 30 m

Also, CE = BC - BE = 90 - 30 = 60 m

Finding the length of DE:

In Δ DEC, by using Pythagoras theorem, we get

$CD^2 = CE^2 + DE^2$

on substituting CD = 100 and CE = 60

$\implies 100^2 = 60^2 + DE^2$

$\implies DE = \sqrt{100^2 - 60^2}$

$\implies DE = \sqrt{6400}$

$\implies \bold{DE = 80 \:m}$

∴ AB = DE = 80 m

Finding the area of Δ DEC:

We know,

$\boxed{\bold{Herons \:Formula:- Area \:of \:a\:triangle = \sqrt{S(S-a)(S-b)(S-c)} }}$

where S = semi-perimeter and a, b & c are the sides of the triangle

In Δ DEC, by using Heron's formula, we get

Semi-Perimeter, $S = \frac{a+b+c}{2} = \frac{100 + 60 + 80}{2} = \frac{240}{2} = 120$

∴ Area of the Δ DEC,

= $\sqrt{120(120-100)(120-60)(120-80)} }}$

= $\sqrt{120\times 20 \times 60 \times 40}$

= $\bold{2400 \:m^2}$

Finding the area of rectangle ABED:

The area of rectangle ABED is,

= Area of a rectangle

= length × breadth

= AD × AB

= 30 × 80

= 2400 m²

Finding the area of trapezium-shaped field:

The area of the trapezium-shaped field ABCD is,

= [Area of ABED] + [Area of DEC]

= 2400 m² + 2400 m²

= 4800 m²

Finding the cost of ploughing:

If the cost of ploughing 1 m² of the field = Rs. 4

Then,

The cost of ploughing 4800 m² of the field will be,

= Rs. 4/m² × 4800 m²

= Rs. 19200

Thus, the total cost of ploughing the trapezium-shaped field is →

Rs. 19200.

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