Question

The base of a triangular field is three times the altitude. If the cost of cultivating the field at $Rs\,24.68$ per hectare be $Rs{\kern 1pt} 333.18$. find base and height.

Answer 1

Hint: First, we need to know about the concept of converting the hectare to meter square, which is to multiply by the number $10000$ by the hectare to make it as a meter value.
Area of the given field can be calculated using the division for the total cost and the rate given.

Complete step-by-step solution:
Since, the base of a triangular field is three times the altitude. Now assume the unknown altitude as a variable $x$ then we get the base of this altitude as $3x$ because it is three times the given altitude.
Now to find the Area of the field we divide the given total cost by the rate, where the total cost is $Rs{\kern 1pt} 333.18$ and the rate is given as $Rs\,24.68$
Hence, we have Area of field $ = \dfrac{{Total{\kern 1pt} {\kern 1pt} \cos t}}{{Rate}}$ and thus applying the values we get
Area of filed $ = \dfrac{{333.18}}{{24.68}} \Rightarrow 13.5{\kern 1pt} {\kern 1pt} {\kern 1pt} hectares$
Now to convert the hectares into meters square, we have $13.5{\kern 1pt} {\kern 1pt} {\kern 1pt} hectares \Rightarrow (13.5 \times 10000){m^2}$
Hence, we get Area of field $ = 135000{m^2}$
Since we know that Altitude $ = x(meters)$ and Base $ = 3x(meter)$
Using the formula to find the altitude of the given problem, which is $\dfrac{1}{2} \times A \times B = \text{Area of field}$ where A is the altitude and B is the base
Thus, substituting the values, we get $\dfrac{1}{2} \times x \times 3x = 13500$
Further solving we have $\dfrac{{3{x^2}}}{2} = 13500$
Using the multiplication and division we get ${x^2} = \dfrac{{13500 \times 2}}{3} \Rightarrow 90000{m^2}$
Taking the square root common, we get $x = 300m$ which is the altitude
And the base is $3x = 3 \times (300m) = 900m$
Therefore, the base is $900m$ and the altitude is $300m$ which is the required answer.

Note: We used the formula of the area of the triangle to find the unknown altitude and base, which is the Area of the triangle equals to $\dfrac{1}{2}bh$
Since while taking out the square root terms like meter square we get $\sqrt {{m^2}} = m$ which is used in the above solution and be careful while solving this.