Suppose that a substance A is converted to substance B at a rate that is proportional to the cube of the amount of B present. The amount of A and B together is always constant, say M. If f( t) = y is the amount of A present at time t, then which of the following differential equation describes the situation? A) $y ^ { \prime }$ = $k ( M - y ) ^ { 3 }$ ; k 0 C) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k 0 D) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k

Question

Suppose that a substance A is converted to substance B at a rate that is proportional to the cube of the amount of B present. The amount of A and B together is always constant, say M. If f( t) = y is the amount of A present at time t, then which of the following differential equation describes the situation? A) $y ^ { \prime }$ = $k ( M - y ) ^ { 3 }$ ; k < 0 B) $y ^ { \prime }$ = $k ( M - y ) ^ { 3 }$ ; k > 0 C) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k > 0 D) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k < 0 E) none of these

Quizplus · Accepted Answer

The rate of conversion of A to B is proportional to the cube of the amount of B present. Since the total amount of A and B is constant (M), the amount of B at any time can be represented as $M - y$. The rate of change of A ($y'$) is negative because A is being converted to B, hence $k < 0$. Therefore, the correct differential equation is $y' = k(M - y)^3$ with $k < 0$.

Suppose That a Substance a Is Converted to Substance B y′y ^ { \prime }y′

Suppose That a Substance a Is Converted to Substance B $y ^ { \prime }$