These are equations, Calculus-style. From modeling real-world phenomenon, from the path of a rocket to the cooling of a physical object, Differential Equations are all around us.

Let \(P(t)\) represent the amount of chemical a factory produces as a function of time \(t\) (in hours). The rate of change of chemical production satisfies the differential equation \[P'(t) = -\ln 3 \times P(t) \left( 1 - \frac{P(t)}{3}\right).\] If the factory alarm is raised when chemical production exceeds \(4\) in \(4\) hours, which of the following inequalities represents the maximum initial amount \(P(0)\) of chemical that guarantees the alarm will not be raised?

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!

Suppose the number of cells in a culture is approximated by \(P(t)\) at time \(t.\) If \(P(t)\) satisfies the differential equation \[P'(t) = \ln 2 \times P(t) \left( 1 - \frac{P(t)}{12}\right)\] and the initial number of cells is \(P(0)=9,\) what is the approximation for the number of cells in the culture at time \(t=3?\)

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!

Suppose the ratio of healthy cells to infected cells in a petri dish at time \(t\) is represented by \(P(t)\). If \(P(t)\) satisfies the logistic differential equation \[P'(t) = -\ln 5 \times P(t) \left( 1 - \frac{P(t)}{7}\right)\]
and \(P(0)= 2\), what is the value of \(P(3)?\)

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!

Suppose the percentage of people surviving a dangerous virus at time \(t\) is approximated by \(P(t)\), where \(P(0)=100.\) If \(P(t)\) satisfies the logistic differential equation \[P'(t) = 6 \times P(t) \left( 1 - \frac{P(t)}{20}\right),\] at what value of \(t\) is the survival rate \(75\)%?

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!

Suppose the population in a park at time \(t\) is given by \(P(t)\), where \[P'(t) = -0.5 \times P(t) \left( 1 - \frac{P(t)}{5}\right).\] If \(P(0) = 12,\) at what time \(t\) does the population in the park reach \(16?\)

Excel in math and science

Master concepts by solving fun, challenging problems.

It's hard to learn from lectures and videos

Learn more effectively through short, conceptual quizzes.

Our wiki is made for math and science

Master advanced concepts through explanations,
examples, and problems from the community.

Used and loved by 4 million people

Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.

Your answer seems reasonable.
Find out if you're right!