Question

The base of a triangular field is three times its height. If the cost of cultivating the field at 2800 per hectare is 37800, find its base and height.
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Answer 1

♣ɢɪᴠᴇɴ:–

• Total cost of cultivating the field = ₹37800
• Rate of cultivating the field = ₹2800/hectare

♣ᴛᴏ ғɪɴᴅ:–

• The base and height of the triangular field = ?

♣sᴏʟᴜᴛɪᴏɴ:–

$\qquad \quad \bull \:$First let us find the area of the triangular field:–

Area of the field:–

$\qquad \quad \bull \quad \sf{ \bigg \lgroup \dfrac{total \: cost}{rate \: per \: hectare} \bigg \rgroup}$

$\qquad \quad : \longrightarrow \tt{ \bigg \lgroup\dfrac{37800}{2800} \bigg \rgroup{hectares}}$

$\qquad \quad : \longrightarrow \tt{ \bigg \lgroup\dfrac{27}{2} \bigg \rgroup{hectares}}$

$\qquad \quad : \longrightarrow \tt{ \bigg \lgroup\dfrac{27}{2} \times 10000\bigg \rgroup{ {m}^{2} }}$

$\qquad \quad : \leadsto \tt {\underline{\boxed {\tt { 135000 { {m}^{2} }}}}}$

Let the height of the field = (x) m

Then it's base = (3x) m

$\qquad \quad \therefore \quad \sf{ \dfrac{1}{2} \times 3x \times x = 135000}$

$\qquad \quad : \longrightarrow \tt{ \dfrac{3 {x}^{2} }{2} = 135000 }$

$\qquad \quad : \longrightarrow \tt{ {x}^{2} = \bigg \lgroup135000 \times \dfrac{2}{3} \bigg \rgroup }$

$\qquad \quad : \longrightarrow \tt{ {x}^{2} = 90000 }$

$\qquad \quad : \longrightarrow \tt{ {x}= \sqrt{90000} }$

$\qquad \quad : \longrightarrow \underline{\boxed{\tt{ {x}= 300 } }}$

$\large \therefore \sf{Height \: of \: the ~field =( x) m = {\underline{\underline{300 \: m}}}}$

$\large \therefore \sf{Base \: of \: the ~field =( 3x) m = (3 \times 300) \: m = {\underline{\underline{900 \: m}}}}$

♣ɴᴏᴛᴇ:–

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