Question

the base of triangular field is three times it's altitudes. if the cost of sowing the field at Rs 58 per hectares is Rs783 then it's base is
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Answer 1

✬ Base = 900 m ✬

Step-by-step explanation:

Given:

• Base of a triangular field is 3 times its altitude.
• Cost of sowing the field is Rs 58 per hectares is Rs 783.

To Find:

• What is the measure of base of field ?

Solution: Let the measure of altitude of field be x . Therefore,

➼ Base of field = 3x

As we know that

★ Area of ∆ = 1/2(Base)(Height) ★

$\implies{\rm }$ Area = 1/2(3x)(x)

$\implies{\rm }$ Area = 3x²/2

Now, cost of sowing the field per hectares is Rs 783.

➟ Total cost = Ar. ∆ $\times$ Cost per hectare

➟ 783 = 3x²/2(58)

➟ 783 = 3x²(29)

➟ 783 = 87x²

➟ 783/87 = x²

➟ 9 = x²

➟ √9 = x

➟ 3 = x

Hence, measure of altitude of triangle is 300m so measure of base will be 3(3) = 900m. [ Changed to metre ]

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[ Another method ]

➼ Area of field from above solution = 3x²/2.......(1)

But also

➨ Area of field = Total cost/Cost per hectares

➨ Area = 783/58

➨ Area = 13.5 hectares

• Since, 1 hectare = 1000 m²

∴ 13.5 hectares = 13.5(1000) = 135000 m².......(2)

A/q

• Equation 1 = Equation 2

$\implies{\rm }$ 135000 = 3x²/2

$\implies{\rm }$ 135000(2) = 3x²

$\implies{\rm }$ 270000 = 3x²

$\implies{\rm }$ 270000/3 = x²

$\implies{\rm }$ √90000 = x

$\implies{\rm }$ 300 = x

So, altitude is x = 300 m and base is 3 times i.e 900m.

Answer 2

Solution :

Let the altitude of be x and then base will be 3x.

________________________

$\dashrightarrow\tt\:\:Area_{{\tiny\triangle}}=\dfrac{1}{2} \times Base \times Height$

$\dashrightarrow\tt\:\:Area_{{\tiny\triangle }} =\dfrac{1}{2} \times 3x \times x$

$\dashrightarrow\:\:\underline{\boxed{\tt Area_{{\tiny\triangle}} = \dfrac{ {3x}^{2} }{2} }}$

_______________________

As we know that,

$\bigstar \: \sf 1 \: hectare = 1000 \: sq.m \: \bigstar$

It is given that,

$\: \: \: \: \: \: \: \: \:\: \tiny \dag \: \sf Cost \: of \: sowing \: of \: the \: field \: per \: hectare \: = \: Rs. \: 58 \: \: \dag$

$\: \: \: \: \: \: \: \: \: \: \tiny \dag \: \sf Total \: rate \: of \: sowing \: the \: field \: = \: Rs. \: 783 \: \dag$

Now, we can find the total cost :

$\bigstar \: \pink{ \underline{ \boxed{ \sf Total \: cost = Area of \: the \: field \times Rs. \: 58}}} \: \bigstar</p><p>$

$: \implies \sf 783 \: = \: \dfrac{3x^2}{2} \: \times \: \dfrac{58}{10000}$

$:\implies \sf x^2 = \dfrac{783}{58} \times \dfrac{2}{3} \times 10000$

$:\implies \sf x^2 = 90000$

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

$:\implies \underline{ \boxed{\sf x = 300 \: m}}$

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Therefore,

$\bullet\:\:\textsf{ Height = \textbf{ 300 \: meter}}$

$\bullet\:\:\textsf{ Base = 3x = 3(300) = \textbf{ 900 \: meter }}$