Question

The base of a rectangular field is three times altitude. if the cost of sowing the field at ₹960 per hectare is ₹12960, find its base and height.​

Answer 1

☆ Correct Question

The base of a triangular field is three times altitude. if the cost of sowing the field at ₹960 per hectare is ₹12960, find its base and height.

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$\bf{ \green{\underline{\underline{ Solutions :}}}} \\ \\ \sf{Total \: cost \: of \: sowing \: the \: field = Rs. \: 12960} \\ \\ \sf{Rate \:per \: hectare = Rs. \: 960 } \\ \\ \sf{Area = \frac{total \: cost}{rate} = \left(\frac{12960}{960} \right)\: hectare } \\ \\ \implies \sf13.5 \: hectares \\ \\ \sf{ \implies (13.5 \times 10000) {m}^{2} } \\ \\ \implies \sf{135000} \: {m}^{2} \\ \\ \sf{Let \: the \: altitude \: of \: the \: field \: be \: x \: meters.} \\ \\ \sf{Then \: its \: base = 3x \: meters.} \\ \\ \sf{Area = \left( \frac{1}{2} \times base \times height \right)} \\ \\ \sf{ \implies \left( \frac{1}{2} \times 3x \times x \right) } {m}^{2} \\ \\ \sf{ \implies \left( \frac{3 {x}^{2} }{2} \right)} {m}^{2} \\ \\ \sf{ \therefore \: \frac{3 {x}^{2} }{2} } = 135000 \\ \\ \sf{ \implies {x}^{2} } = \left(135000 \times \frac{2}{3} \right) \\ \\ \implies \sf{90000} \\ \\ \implies\sf x = \sqrt{90000} = 300 \\ \\ \sf{\purple{\therefore \: base = (3 \times 300)m = 900 \: m \: and \:}} \\ \sf{ \purple{altitude = 300 \: m}}$

Answer 2

Given : Base of a triangular field is 3 times the altitude & Cost of sowing field at Rs 960 is Rs 12960.

To Find : Find the meaure of base and height ?

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Solution : Let the altitude of field be x.

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Here, Total cost of sowing field is Rs 12960 and rate per hectare is Rs 960.

• $\leadsto$Base of field = 3(x)

$\qquad{\sf:\implies{Area~=~\dfrac{Total~cost}{Rate}}}$

$\qquad{\sf:\implies{\Bigg(\dfrac{12960}{960}\Bigg)~hectares}}$

$\qquad{\sf:\implies{13.5~hectares}}$

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• 1 hectare = 10000 m² so
• 13.5 hectares = 13.5(10000)m²

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◗Total area of field = 135000 m²

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$\underline{\frak{As~ we ~know~ that~:}}$

• $\boxed{\sf\pink{Area~of~Triangle_{\triangle}~=~\dfrac{1}{2}\bigg(Base\bigg)\bigg(Height\bigg)}}$

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$\qquad{\sf:\implies{135000~=~\dfrac{1}{2}\bigg(3x\bigg)\bigg(x\bigg)}}$

$\qquad{\sf:\implies{135000~=~\dfrac{3x^2}{2}}}$

$\qquad{\sf:\implies{x^2~=~135000\bigg(2\bigg)}}$

$\qquad{\sf:\implies{x^2~=~\cancel\dfrac{270000}{3}}}$

$\qquad{\sf:\implies{x^2~=~90000~}}$

$\qquad{\sf:\implies{x~=~\sqrt{90000}}}$

$\qquad:\implies{\underline{\boxed{\frak{\pink{x~=~300}}}}}$★

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Hence,

• $\underset{\blue {\bf Required\ Answer}}{\underbrace{\boxed{\frak{\pink{Measure~of~\dfrac{Altitude~is~x~=~300m}{Base~is~3x~=~3(300)~=~900m}}}}}}$