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Practice problems on Differential Equations

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Practice problems on Differential Equations

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Differential Equations Quiz 1

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Practice problems on Differential Equations

"},"featured_media":0,"featured_media_url":"$undefined","acf":{"duree_estimee":15,"niveau_difficulte":"$undefined","categorie":"Differential Equations","nombre_questions":10,"score_minimum":70,"ordre_questions":"Fixe","questions":[{"id":"cmnga4qxe000al4040cjfu8si","texte_question":"

A culture of bacteria grows at a rate proportional to the number present. Initially, there are $$20$$ bacteria. After $$10$$ minutes, the population grows to $$60$$ bacteria. How many bacteria will there be after $$30$$ minutes?

","type_question":"QCM","explication":"

Exponential growth model: $$P(t)=P_0e^{kt}$$ Given: $$P(0)=20$$ $$P(10)=60$$ Solve for $$k$$: $$60=20e^{10k}$$ $$3=e^{10k}$$ $$k=\\frac{\\ln3}{10}$$ Now find $$P(30)$$: $$P(30)=20e^{30k}$$ $$=20e^{3\\ln3}$$ $$=20\\cdot27$$ $$=540$$ Final answer: $$540$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$180$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"

$$540$$

","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"},{"texte":"

$$120$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"

$$360$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]},{"id":"cmnga4qxe000fl404rce68i3w","texte_question":"

A radioactive isotope has a half-life of $$20$$ days. If a sample initially has a mass of $$5$$ kilograms, how long will it take for the sample to decay to $$200$$ grams?

","type_question":"QCM","explication":"

Convert units: $$5\\text{ kg}=5000\\text{ g}$$ Exponential decay: $$M(t)=M_0\\left(\\frac12\\right)^{t/20}$$ Set $$200=5000\\left(\\frac12\\right)^{t/20}$$ $$\\frac{200}{5000}=\\left(\\frac12\\right)^{t/20}$$ $$\\frac1{25}=\\left(\\frac12\\right)^{t/20}$$ Take logs: $$\\ln\\left(\\frac1{25}\\right)=\\frac{t}{20}\\ln\\left(\\frac12\\right)$$ Solve: $$t=20\\frac{\\ln(1/25)}{\\ln(1/2)}$$ $$t\\approx92.9$$ Final answer: $$92\\text{ days (approx)}$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$86\\text{ days}$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"

$$92\\text{ days}$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"},{"texte":"

$$100\\text{ days}$$

","correcte":true,"explication":"

Correct

","imageUrl":"$undefined"},{"texte":"

$$110\\text{ days}$$

","correcte":false,"explication":"

Incorrect

","imageUrl":"$undefined"}]}]},"categories":["Differential Equations"],"date":"2026-04-01T15:56:41.040Z","modified":"2026-04-01T16:48:00.338Z"}}]]}]}]]