
The base of the prism is triangular in shape with sides of 3 cm, 4 cm, 5 cm. Find the volume of the prism if it’s height is 10 cm.
a) 45 cu.cm
b) 50 cu.cm
c) 60 cu.cm
d) 65 cu.cm
Answer
512.7k+ 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
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
What constitutes the central nervous system How are class 10 biology CBSE

Explain the Treaty of Vienna of 1815 class 10 social science CBSE

Imagine that you have the opportunity to interview class 10 english CBSE

This cake is sweet that one A As sweet as B Sweeter class 10 english CBSE

Compare the advantages and disadvantages of multipurpose class 10 social science CBSE

Lets have a cup of tea A Havent we B Have we C Will class 10 english CBSE
