In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is l. The altitude of the prism for which the volume is greatest :

Question

In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is l. The altitude of the prism for which the volume is greatest : (A) (l/2) (B) (l/√3) (C) (l/3) (D) (l/4)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/41da1e180ee83e69d7ae476aef695523-.png" /> AG =(2/3) ⋅ ( a √3/2)=( a /√3). Now l2=( a 2/3)+ h 2 ; operatornamearca( A )=(√3/4) a 2 V ( h )=(√3/4) a 2 h =(√3/4) h (l2- h 2) 3 ; V prime( h )=0 ⇒ h =(l/√3) ; V max =(l3/2)

Q. In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is $l$ . The altitude of the prism for which the volume is greatest :

$\frac{l}{2}$

$\frac{l}{3}$

$\frac{l}{3}$

$\frac{l}{4}$

$A G = \frac{2}{3} \cdot \frac{a 3}{2} = \frac{a}{3}$ . Now $l^{2} = \frac{a ^{2}}{3} + h^{2}; arca (A) = \frac{3}{4} a^{2}$
$V (h) = \frac{3}{4} a^{2} h = \frac{3}{4} h (l^{2} - h^{2}) 3; V^{'} (h) = 0 \Rightarrow h = \frac{l}{3}; V_{m a x} = \frac{l ^{3}}{2}$

Q. In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is l. The altitude of the prism for which the volume is greatest :

2l​

3​l​

3l​

4l​

AG=32​⋅2a3​​=3​a​. Now l2=3a2​+h2;arca(A)=43​​a2 V(h)=43​​a2h=43​​h(l2−h2)3;V′(h)=0⇒h=3​l​;Vmax​=2l3​

Q. In a regular triangular prism the distance from the centre of one base to one of the vertices of the other base is $l$ . The altitude of the prism for which the volume is greatest :

$\frac{l}{2}$

$\frac{l}{3}$

$\frac{l}{3}$

$\frac{l}{4}$

$A G = \frac{2}{3} \cdot \frac{a 3}{2} = \frac{a}{3}$ . Now $l^{2} = \frac{a ^{2}}{3} + h^{2}; arca (A) = \frac{3}{4} a^{2}$
$V (h) = \frac{3}{4} a^{2} h = \frac{3}{4} h (l^{2} - h^{2}) 3; V^{'} (h) = 0 \Rightarrow h = \frac{l}{3}; V_{m a x} = \frac{l ^{3}}{2}$