# The sides of the triangular board are 13 metres, 14 metres, 15metres. The cost of painting it at the rate of Rs. 8.75\[{{m}^{2}}\] per is?

a) Rs. 688.80

b) Rs. 735

c) Rs. 730.80

d) Rs. 722.50

Last updated date: 18th Mar 2023

•

Total views: 303.6k

•

Views today: 8.83k

Answer

Verified

303.6k+ views

Hint: To solve the question, we have to calculate the area of the given triangular board to calculate the portion of area to be painted. The obtained when is multiplied with the rate of cost of painting, will give you the total cost of painting the triangular board.

Complete step-by-step answer:

We know that the portion of area to be painted is equal to the area of the triangular board.

To calculate the area of the triangular, we use the Heron's formula of area of a triangle which is equal to \[\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]

Where a, b, c are three sides of the triangle and s is the semi-perimeter of the triangle.

We know that the formula for semi-perimeter of triangle with sides a, b, c is given by \[s=\dfrac{a+b+c}{2}\]

The given sides of the triangular board are 13 metres, 14 metres, 15metres. By substituting the values of sides of triangle in the above mentioned formula of semi-perimeter, we get

\[s=\dfrac{13+14+15}{2}=\dfrac{42}{2}\]

\[\Rightarrow s=21\]

By substituting the values of semi-perimeter and the sides of triangle in the above mentioned formula of area of a triangular board, we get

\[A=\sqrt{21\left( 21-13 \right)\left( 21-14 \right)\left( 21-15 \right)}\]

Where A represents the area of a triangular board.

\[A=\sqrt{21\left( 8 \right)\left( 7 \right)\left( 6 \right)}\]

By writing the values in terms of prime numbers we get

\[A=\sqrt{\left( 7\times 3 \right)\left( 2\times 2\times 2 \right)\left( 7 \right)\left( 3\times 2 \right)}\]

By rearranging the values we get

\[A=\sqrt{\left( 7\times 7 \right)\left( 2\times 2\times 2\times 2 \right)\left( 3\times 3 \right)}\]

\[A=\sqrt{{{7}^{2}}\times {{2}^{4}}\times {{3}^{2}}}\]

\[\begin{align}

& A=\sqrt{{{\left( 7\times {{2}^{2}}\times 3 \right)}^{2}}} \\

& =7\times {{2}^{2}}\times 3 \\

& =21\times 4 \\

& =48{{m}^{2}} \\

\end{align}\]

Thus, the area of the triangular board of sides 13 metres, 14 metres, 15metres is equal to 48\[{{m}^{2}}\]

The cost of painting the triangular board at the rate of Rs. 8.75 per\[{{m}^{2}}\] = 8.75 \[\times \] (area of the triangular board of given sides).

The cost of painting the triangular board at the rate of Rs. 8.75 per\[{{m}^{2}}\] = \[8.75\times 48\] = Rs. 420.

Note: The possibility of mistake can be not able to analyse the need to calculate the area of the triangled board to calculate the total cost of painting of the triangular board of given sides. The alternative way to calculate the area of triangle of sides a, b, c is to use the formula \[\dfrac{1}{2}ab\sin C\] where \[\sin C\] can be calculated by using\[\cos C=\dfrac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab},\cos {{C}^{2}}+\sin {{C}^{2}}=1\].

Complete step-by-step answer:

We know that the portion of area to be painted is equal to the area of the triangular board.

To calculate the area of the triangular, we use the Heron's formula of area of a triangle which is equal to \[\sqrt{s\left( s-a \right)\left( s-b \right)\left( s-c \right)}\]

Where a, b, c are three sides of the triangle and s is the semi-perimeter of the triangle.

We know that the formula for semi-perimeter of triangle with sides a, b, c is given by \[s=\dfrac{a+b+c}{2}\]

The given sides of the triangular board are 13 metres, 14 metres, 15metres. By substituting the values of sides of triangle in the above mentioned formula of semi-perimeter, we get

\[s=\dfrac{13+14+15}{2}=\dfrac{42}{2}\]

\[\Rightarrow s=21\]

By substituting the values of semi-perimeter and the sides of triangle in the above mentioned formula of area of a triangular board, we get

\[A=\sqrt{21\left( 21-13 \right)\left( 21-14 \right)\left( 21-15 \right)}\]

Where A represents the area of a triangular board.

\[A=\sqrt{21\left( 8 \right)\left( 7 \right)\left( 6 \right)}\]

By writing the values in terms of prime numbers we get

\[A=\sqrt{\left( 7\times 3 \right)\left( 2\times 2\times 2 \right)\left( 7 \right)\left( 3\times 2 \right)}\]

By rearranging the values we get

\[A=\sqrt{\left( 7\times 7 \right)\left( 2\times 2\times 2\times 2 \right)\left( 3\times 3 \right)}\]

\[A=\sqrt{{{7}^{2}}\times {{2}^{4}}\times {{3}^{2}}}\]

\[\begin{align}

& A=\sqrt{{{\left( 7\times {{2}^{2}}\times 3 \right)}^{2}}} \\

& =7\times {{2}^{2}}\times 3 \\

& =21\times 4 \\

& =48{{m}^{2}} \\

\end{align}\]

Thus, the area of the triangular board of sides 13 metres, 14 metres, 15metres is equal to 48\[{{m}^{2}}\]

The cost of painting the triangular board at the rate of Rs. 8.75 per\[{{m}^{2}}\] = 8.75 \[\times \] (area of the triangular board of given sides).

The cost of painting the triangular board at the rate of Rs. 8.75 per\[{{m}^{2}}\] = \[8.75\times 48\] = Rs. 420.

Note: The possibility of mistake can be not able to analyse the need to calculate the area of the triangled board to calculate the total cost of painting of the triangular board of given sides. The alternative way to calculate the area of triangle of sides a, b, c is to use the formula \[\dfrac{1}{2}ab\sin C\] where \[\sin C\] can be calculated by using\[\cos C=\dfrac{{{a}^{2}}+{{b}^{2}}-{{c}^{2}}}{2ab},\cos {{C}^{2}}+\sin {{C}^{2}}=1\].

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India