A solid rectangular brick is to be made from 1 cu feet. of clay. The brick must be 3 times as long as it is wide. The ratio of height to width of the brick for which it will have minimum surface area, is

Question

A solid rectangular brick is to be made from 1 cu feet. of clay. The brick must be 3 times as long as it is wide. The ratio of height to width of the brick for which it will have minimum surface area, is (A) (2/9) (B) ((2/9))(1/3) (C) (2/3) (D) (3/2)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/8622d3d2340e4cbdefd59da3fdac9ef7-.png" /> 3 x2 h=1 S=2(h x+3 x2+3 x h) S=2(3 x2+4 h x)=2(3 x2+4 x ⋅ (1/3 x2)) S(x)=2(3 x2+(4/3 x)) S prime(x)=2(6 x+(4/3 x2))=0 ; x3=(2/9) ⇒ x=((2/9))(1/3) now h =(1/3 x 2) ; hence ( h / x )=(1/3 x 2) ⋅ (1/ x )=(1/3 x 3)=(1 ⋅ 9/3 ⋅ 2)=(3/2)

Q. A solid rectangular brick is to be made from 1 cu feet. of clay. The brick must be 3 times as long as it is wide. The ratio of height to width of the brick for which it will have minimum surface area, is

$\frac{2}{9}$

$(\frac{2}{9})^{\frac{1}{3}}$

$\frac{2}{3}$

$\frac{3}{2}$

Q. A solid rectangular brick is to be made from 1 cu feet. of clay. The brick must be 3 times as long as it is wide. The ratio of height to width of the brick for which it will have minimum surface area, is

92​

(92​)31​

32​

23​

3x2h=1 S=2(hx+3x2+3xh) S=2(3x2+4hx)=2(3x2+4x⋅3x21​) S(x)=2(3x2+3x4​) S′(x)=2(6x+3x24​)=0;x3=92​ ⇒x=(92​)31​ now h=3x21​; hence xh​=3x21​⋅x1​=3x31​=3⋅21⋅9​=23​

$\frac{2}{9}$

$(\frac{2}{9})^{\frac{1}{3}}$

$\frac{2}{3}$

$\frac{3}{2}$