Question

2. The base of a triangular field is three times its altitude. If the cost of
sowing the field at Rs 58 per hectare is Rs 783, find its base and height.
21am and​

Answer 1

The base and the height of a triangular field :-

• The base of the field = 900 m.
• The height of the field = 300 m.

Given :

• The base of a triangular field is three times it's altitude.
• The cost of sowing the field at Rs. 58 per hectare = Rs. 783.

To Find :

• The base of a triangular field.
• The height of a triangular field.

Solution :

Let,

The height be x.

The base be 3x because according to the question, the base of a triangular field is three times it's altitude.

We know that,

$\huge\blue \star \: \: \: \large{ \boxed{ \bf{ \pink{A = \dfrac{1}{2} \times b \times h }}}}$

Where,

• Area is denoted with A.
• Base is denoted with b.
• Height is denoted with h.

$\bf \implies \dfrac{1}{2} \times3x \times x \\ \\ \\ \bf \implies \dfrac{1}{2} \times {3x}^{2}$

According to the question,

The cost of sowing the field at Rs. 58 per hectare is Rs. 783.

$\bf \implies \dfrac{1}{2} \times {3x}^{2} \times 58 = 783 \: H \\ \\ \\ \bf \implies {3x}^{2} \times 58 = 783 \times 2 \: H \\ \\ \\ \bf \implies {3x}^{2} \times 58 = 1566 \: H \\ \\ \\ \bf \implies {3x}^{2} = \dfrac{1566}{58} \: H \\ \\ \\ \bf \implies {3x}^{2} = 27 \: H \\ \\ \\ \bf \implies {x}^{2} = \dfrac{27}{3} \: H \\ \\ \\ \bf \implies {x}^{2} = 9 \: H$

We know that,

• 1 hectare (H) = 10000 m².

$\bf \implies {x}^{2} = 9 \times 10000 \: {m}^{2} \\ \\ \\ \bf \implies {x}^{2} = 90000 \: {m}^{2} \\ \\ \\ \bf \implies x = \sqrt{90000 \: {m}^{2} } \\ \\ \\ \bf \implies x = 300 \: m \\ \\ \\ \: \: \bf \therefore \: \: Height = 300 \: m$

Therefore,

The height of a triangular field = x = 300 m.

The base of a triangular field = 3x = 3 × 300 m = 900 m.

Hence,

The height and the base of a triangular field is 300 m and 900 m.