Question

The base of a triangular field is 4 times its height. If the cost of cultivating the field at rs 450 per square metre is rs 8100, find its base and height

Answer 1

✡ Given :

✮ Base of the triangular field is 4 times the height.

✮ Cost of cultivating the field per sq. metre is Rs. 450

✮ Cost of cultivating whole field is Rs. 8100

Required to find :

➦ Base and height of the triangle .

Explanation ↩ :

➼ In the question it is given that the base of the triangular field is 4 times the height .

➼So, let's consider the height be x and base be 4x ( this is because it is given in the question) .

➼Here " x " is any integer .

➼ Now , considering the next part of the question we have many hints .

➼ It is given that ;

Cost of cultivating the field per sq. meter is Rs. 450

➼ Similarly;

Cost of cultivating whole field is ;

Y × 450 [ here y is the area of the whole field]

= 8100

➼Now we have to find the area of the whole field ;

☆ This is found by dividing Cost of cultivating whole field by Cost of cultivating per sq. meter .

➼ Hence we will come to know the area of the whole field .

➼ Now using the most familiar formula that is

$\boxed{\large{Area\:of\:the\:triangle\:=\: \frac{1}{2} \times base \times height}}$

➼ we can find the length of base and height .

♧ Now ; Let's crack the required Solution.

Solution ✏ :

Given statement :

⇥Base is 4 times the height .

So, let's consider

Height of the triangularfield be x

Base of the triangular field be 4x

Similarly;

It is also given that,

Cost of cultivating field per square meter is Rs 450 .

Cost of cultivating whole field = Rs. 8100

Using the formula;

$\boxed{\longrightarrow{\boxed{Area\: of\: the\: whole\: field\: =\; \frac{Cost \: of\: cultivating\: whole \:field}{Cost \:of \:cultivating \:per\:square\:meter}}}}$

Hence;

$\longrightarrow{Area \:of\: the \:whole\: field \: =\: \frac{8100}{450}}$

$\longrightarrow{Area \:of\: the \:whole\: field \: =\: 18 {meter}^{2}}$

So;

Area of the whole field is 18 meter^2

However;

We know that,

$\boxed{\large{Area\:of\:the\:triangle\:=\: \frac{1}{2} \times base \times height}}$

So,

Let's substitute the required values ,◇◇

$\Rightarrow{\textsf{Area of the field =}}{\frac{1}{2} \times 4x \times x }$

$\Rightarrow{18{meter}^{2} = \frac{{4x}^{2}}{2}}$

Now transpose 2 to the L.H.S side

$\Rightarrow{18 \times 2 = {4x}^{2}}$

$\Rightarrow{36 = {4x}^{2}}$

Now transpose L.H.S terms to R.H.S and R.H.S terms to L.H.S

$\Rightarrow{{4x}^{2} = 36 }$

$\Rightarrow{{x}^{2} = \frac{36}{4}}$

$\Rightarrow{{x}^{2} = \cancel {\frac{36}{4}}}$

$\Rightarrow{{x}^{2} = 9 }$

$\Rightarrow{ x = \sqrt{9}}$

$\longrightarrow{ x = 3 }$

Now substitute this value in place of x in base and height;

Base = 4x = 4(3) = 12 meters

Height = x = 3 meters.

Therefore;

$\boxed{\longrightarrow{\boxed{Base\:=\:12 meters}}}$

$\boxed{\longrightarrow{\boxed{Height\:=\:3 meters}}}$

✔ Hence Solved ..