Answer:
Base = 900 meters
Height = 300 meters
Step-by-step explanation:
The area of a triangle with base b and height h is given by the formula:
[tex]\textsf{Area of a triangle}=\dfrac{1}{2}bh[/tex]
The altitude of a triangle is its height.
Given that the base (b) is three times the height (h), then b = 3h. Substitute this into the area formula:
[tex]\textsf{Area of triangular field}=\dfrac{1}{2}3h\cdot h\\\\\\\textsf{Area of triangular field}=\dfrac{3}{2}h^2[/tex]
To find the area of the triangular field, we can the total cost of ₹1012.50 by the rate of ₹75 per hectare:
[tex]\textsf{Area of triangular field}=\dfrac{\textsf{Total cost}}{\textsf{Rate}}\\\\\\\textsf{Area of triangular field}=\dfrac{1012.50}{75}\\\\\\\textsf{Area of triangular field}=13.5\; \sf hectares[/tex]
As one hectare is equal to 10,000 m², then the area of the field in square meters is:
[tex]\textsf{Area of triangular field}=135000\; \sf m^2[/tex]
To find the height (altitude) of the triangular field, substitute the formula for area into this equation and solve for h:
[tex]\dfrac{3}{2}h^2=135000\\\\\\\dfrac{3}{2}h^2\cdot \dfrac{2}{3}=135000\cdot \dfrac{2}{3}\\\\\\h^2=90000\\\\\\\sqrt{h^2}=\sqrt{90000}\\\\\\h=300\;\sf meters[/tex]
Therefore, the height of the field is 300 meters.
Since the base is three times the height, then:
[tex]\textsf{Base}=3 \cdot 300 = 900\; \sf meters[/tex]
So, the base is 900 meters.
