Question 1

Which of the following is the general form of a separable differential equation?

Accepted Answer

$$\frac{dy}{dt} = g(t) h(y)$$

Question 2

Given the separable differential equation $$\frac{dy}{dt} = ky$$, what is the general solution?

Accepted Answer

$$y = Ce$$^{kt}$$

Question 3

A radioactive substance decays according to $$m = m_0$ e$$^{-kt}$$. If the half-life is 5730 years, what is the value of k$?

Accepted Answer

$$k = \frac{\ln 2}{5730}$$

Question 4

If $$m_0 = 100$ grams of a radioactive material is present initially, how much remains after one half-life?

Accepted Answer

$$50$$ grams

Question 5

The half-life ($$t_{1/2}) of a $$substance is defined as:

Accepted Answer

The time required for half of the substance to decay.

Question 6

Newton's Law of Cooling states that the rate of change of temperature is proportional to:

Accepted Answer

The difference between the object's temperature and the ambient temperature.

Question 7

Given Newton's Law of Cooling $$\frac{dH}{dt} = -k(H - H_s)$$, what is the solution for $$H if$$ $$H(0) = H_0$?

Accepted Answer

$$H = H_s + (H_0$ - H_s)e$$^{-kt}$$

Question 8

A cup of coffee at $$90^\circC is $$placed in a room at $$18^\circC. $$According to Newton's Law of Cooling, which equation models the temperature $$H at $$time $$t$$?

Accepted Answer

$$H = 18 + 72e$$^{-kt}$$

Question 9

If $$m = m_0$ e$$^{-kt}$$ and m = 0.9$$ $$m_0$$ after $$t$$ years, and $$k = \frac{\ln 2}{5730}$$, what is $$t$$?

Accepted Answer

$$t = \frac{\ln 0.9}{-k}$$

Question 10

Which of the following best describes the graph of radioactive decay $$m = m_0$ e$$^{-kt}$$?

Accepted Answer

It is a decreasing exponential curve approaching zero.

Question 11

A scientist measures that a sample of carbon-14 has decayed to 90% of its original amount. Using $$k = \frac{\ln 2}{5730}$$, how many years have passed?

Accepted Answer

$$t = \frac{\ln 0.9}{-k}$$

Question 12

Which of the following is NOT an application of separable differential equations as discussed in the materials?

Accepted Answer

Projectile motion with air resistance

Question 13

If $$y = 300e$$^{kt}$$ and y(0) = 300$, what is the value of k$ if y(2) = 600$?

Accepted Answer

$$k = \frac{\ln 2}{2}$$

Question 14

Which substitution is used to simplify Newton's Law of Cooling $$\frac{dH}{dt} = -k(H - H_s)$$?

Accepted Answer

$$y = H - H_s$$

Question 15

A household egg at $$90^\circC is $$put in a bath of $$18^\circC $$water. If the temperature after $$t$$ minutes is $$H(t)$$, which equation models this scenario?

Accepted Answer

$$H(t) = 18 + 72e$$^{-kt}$$

7.2 Separable Differential Equations

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