JEECUP (UP Polytechnic) Applications — practice questions

Explore 2 free Applications questions in JEECUP (UP Polytechnic) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Applications questions and exercises with answers

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The half-life of a radioactive substance decaying by $dm/dt = -km$ (with $k > 0$) in terms of $k$ is:Read the answer and worked solution →Newton's law of cooling: $dT/dt = -k(T - T_s)$ where $T_s$ is the surrounding temperature. The general solution is:Read the answer and worked solution →

Worked practice questions

The half-life of a radioactive substance decaying by $dm/dt = -km$ (with $k > 0$) in terms of $k$ is:

$\ln 2/k$
$1/k$
$k \ln 2$
$2/k$

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Correct answer: A. $\ln 2/k$

$m(t) = m_0 e^{-kt}$. **Half-life:** $m_0/2 = m_0 e^{-kt_{1/2}}$ ⇒ $kt_{1/2} = \ln 2$ ⇒ $t_{1/2} = \ln 2/k$.

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Newton's law of cooling: $dT/dt = -k(T - T_s)$ where $T_s$ is the surrounding temperature. The general solution is:

$T = T_s + c e^{-kt}$
$T = T_s + c e^{kt}$
$T = T_s - ct$
$T = T_s t + c$

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Correct answer: A. $T = T_s + c e^{-kt}$

Substituting $u = T - T_s$ gives $du/dt = -ku$ ⇒ $u = c e^{-kt}$ ⇒ $T = T_s + c e^{-kt}$. As $t \to \infty$, $T \to T_s$ (body equilibrates with surroundings).