Answer

Verified

447.9k+ views

Hint: To solve the question, we have to calculate the area of the triangular base of the prism which when is multiplied with the height of the prism, will give you the volume of the prism.

Complete step-by-step answer:

We know that,

The volume of a prism = Area of the base of the prism \[\times \]the height of the prism ..… (1)

To calculate the area of the triangular base of the prism, 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 base of the prism are 3 cm, 4 cm, 5 cm. By substituting the values of sides of triangle in the above mentioned formula of semi-perimeter, we get

\[s=\dfrac{3+4+5}{2}=\dfrac{12}{2}\]

\[\Rightarrow s=6\]

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

\[A=\sqrt{6\left( 6-3 \right)\left( 6-4 \right)\left( 6-5 \right)}\]

Where A represents the area of a triangular base of the prism.

\[A=\sqrt{6\left( 3 \right)\left( 2 \right)\left( 1 \right)}\]

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

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

\[\begin{align}

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

& =2\times 3 \\

& =6c{{m}^{2}} \\

\end{align}\]

Thus, the area of the triangular base of the prism of sides 3 cm, 4 cm, 5 cm is equal to 6\[c{{m}^{2}}\].

The given value of height of the prism is equal to 10 cm.

By substituting the given and calculated values in equation (1) we get

Thus, the volume of a prism \[=6\times 10=60cu.cm\]

Hence, option (c) is the right choice.

Note: The possibility of mistake can be, not able to apply the correct formula for volume of prism and for the area of triangles of given sides. The alternative way of calculating the area of the triangle is by using the formula \[\dfrac{1}{2}bh\] where b, h are the base and height of the triangle. Since the given sides of the triangle from the right-angle triangle, the other sides of the triangle excluding hypotenuse, form the base and height of the triangle. Thus, we can calculate the area of the triangular base of the prism.

Complete step-by-step answer:

We know that,

The volume of a prism = Area of the base of the prism \[\times \]the height of the prism ..… (1)

To calculate the area of the triangular base of the prism, 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 base of the prism are 3 cm, 4 cm, 5 cm. By substituting the values of sides of triangle in the above mentioned formula of semi-perimeter, we get

\[s=\dfrac{3+4+5}{2}=\dfrac{12}{2}\]

\[\Rightarrow s=6\]

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

\[A=\sqrt{6\left( 6-3 \right)\left( 6-4 \right)\left( 6-5 \right)}\]

Where A represents the area of a triangular base of the prism.

\[A=\sqrt{6\left( 3 \right)\left( 2 \right)\left( 1 \right)}\]

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

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

\[\begin{align}

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

& =2\times 3 \\

& =6c{{m}^{2}} \\

\end{align}\]

Thus, the area of the triangular base of the prism of sides 3 cm, 4 cm, 5 cm is equal to 6\[c{{m}^{2}}\].

The given value of height of the prism is equal to 10 cm.

By substituting the given and calculated values in equation (1) we get

Thus, the volume of a prism \[=6\times 10=60cu.cm\]

Hence, option (c) is the right choice.

Note: The possibility of mistake can be, not able to apply the correct formula for volume of prism and for the area of triangles of given sides. The alternative way of calculating the area of the triangle is by using the formula \[\dfrac{1}{2}bh\] where b, h are the base and height of the triangle. Since the given sides of the triangle from the right-angle triangle, the other sides of the triangle excluding hypotenuse, form the base and height of the triangle. Thus, we can calculate the area of the triangular base of the prism.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The 3 + 3 times 3 3 + 3 What is the right answer and class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How many crores make 10 million class 7 maths CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

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