Question 606143
Let the volume of the smaller cube = {{{ V[1] }}}
Let the volume of the bigger cube = {{{ V[2] }}}
Let the side of the smaller cube = {{{ s[1] }}}
Let the side of the bigger cube = {{{ s[2] }}}
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given:
{{{ s[1]/s[2] = 2/3 }}}
(1) {{{ s[1] = (2/3)*s[2] }}}
(2) {{{ V[2] - V[1] = 152 }}}
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The formulas for volume are
{{{ V[1] = s[1]^3 }}}
{{{ V[2] = s[2]^3 }}}
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Substituting into (2),
(2) {{{ V[2] - V[1] = 152 }}}
(2) {{{ s[2]^3 - s[1]^3 = 152 }}}
Substitue (1) into (2)
(2) {{{ s[2]^3 - ((2/3)*s[2])^3 = 152 }}}
(2) {{{ s[2]^3 - (8/27)*s[2]^3 = 152 }}}
(2) {{{ s[2]^3*( 1 - 8/27 ) = 152 }}}
(2) {{{ s[2]^3*( 19/27 ) = 152 }}}
(2) {{{ s[2]^3 = (27/19)*152 }}}
(2) {{{ s[2]^3 = 216 }}}
(2) {{{ s[2] = root(3,216) }}}
(2) {{{ s[2] = 6 }}}
and, since
(1) {{{ s[1] = (2/3)*s[2] }}}
(1) {{{ s[1] = (2/3)*6 }}}
(1) {{{ s[1] = 4 }}}
The side of the bigger cube is 6
check:
{{{ V[1] = 4^3 }}}
{{{ V[2] = 6^3 }}}
and
(2) {{{ V[2] - V[1] = 152 }}}
(2) {{{ 6^3 - 4^3 = 152 }}}
(2) {{{ 216 - 64 = 152 }}}
(2) {{{ 152 = 152 }}}
OK