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Calculus 1 : How to find constant of proportionality of rate

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Example Question #1 : How To Find Constant Of Proportionality Of Rate

Suppose a blood cell increases proportionally to the present amount. If there were blood cells to begin with, and blood cells are present after hours, what is the growth constant?

Possible Answers:

$\frac{3}{2}$

$\frac{2}{3}$

$\frac{1}{2}ln\left(\frac{3}{2}\right)$

$ln\left(\frac{3}{2}\right)$

$ln\left(\frac{2}{3}\right)$

Correct answer:

$\frac{1}{2}ln\left(\frac{3}{2}\right)$

Explanation:

The population size P(t) after some time is given by:

P(t)= P_ie^k^t

where P_i is the initial population.

At the start, there were 30 blood cells.

P_i=P(0)=30

Substitute this value into the given formula.

P(t)= 30e^k^t

After 2 hours, 45 blood cells were present. Write this in mathematical form.

P(2)=45

Substitute this into P(t)= 30e^k^t , and solve for .

45= 30e^k^(^2^)

$\frac{3}{2}= e^2^k$

$ln\left(\frac{3}{2}\right)=ln(e^2^k)$

$2k=ln\left(\frac{3}{2}\right)$

$k=\frac{1}{2}ln\left(\frac{3}{2}\right)$

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Example Question #1 : Constant Of Proportionality

Given any linear function y=mx+b , determine the direct constant of proportionality

Possible Answers:

$\frac{m}{b}$

$\frac{m-b}{m}$

m/b

Correct answer:

Explanation:

Direct constant of proportionality for any given function y, between any x values, is given by

$r=\frac{\Delta y}{\Delta x}$ , where is the direction constant of proportionality

In the case of a linear function

$\frac{\Delta y}{\Delta x}$ is the same thing as the slope.

Therefore, the constant of proportionality is

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Example Question #2 : Constant Of Proportionality

Find the direct constant of proportionality of x^2 from to 2.5 .

Possible Answers:

3.5

2.5

Correct answer:

3.5

Explanation:

To determine the direct constant of proportionality, we determine the rate of change from x=1 and x=2.5 for x^2 .

Rate of change is determined by

$r=\frac{\Delta y}{\Delta x}$ .

In our case, y=x^2 between x=1 and x=2.5 , the rate of change is

$r=\frac{2.5^2-1}{2.5-1}=\frac{5.25}{1.5}=3.5$ .

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Example Question #1 : Constant Of Proportionality

Find the direct constant of proportionality of y=ln(3x) from x=2 to x=4 .

Possible Answers:

r= ln(12)-ln(6)

$r= \frac{6}{2ln(2)}$

$r= \frac{ln(2)}{2}$

$r= \frac{2}{ln(2)}$

Correct answer:

$r= \frac{ln(2)}{2}$

Explanation:

Direct constant of proportionality is given by

$r=\frac{\Delta y}{\Delta x}$ .

Since y=ln(3x) and $\Delta x= 4-2=2$

$r= \frac{\Delta y}{\Delta x}=\frac{ln(3*4)-ln(3*2)}{2}= \frac{ln(2)}{2}$

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Example Question #2 : How To Find Constant Of Proportionality Of Rate

Suppose a population of bacteria increases from 100 to 1000 in $30\: min$ . What is the constant of growth?

Possible Answers:

$e^{30}$

$\ln(10)$

$\frac{\ln(10)}{30}$

None of these

Correct answer:

$\frac{\ln(10)}{30}$

Explanation:

The equation for population growth is given by $P=Ae^{kt}$ . is the population, is the intial value, is time, and is the growth constant. We can plug in the values we know at time t=0 and solve for .

$100=Ae^{0k}=A$

Now that we solved for , we can plug in what we know for time t=30 and solve for .

$1000=100e^{30k}$

$e^{30k}=10$

$30k=\ln(10)$

$k=\frac{\ln(10)}{30}$

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Example Question #1 : Constant Of Proportionality

A population of deer grew from 50 to 200 in 7 years. What is the growth constant for this population?

Possible Answers:

$\frac{\ln(4)}{7}$

$\frac{4}{7}$

$\ln(4)$

None of these

$\ln(7)$

Correct answer:

$\frac{\ln(4)}{7}$

Explanation:

The equation for population growth is given by $P=Ae^{kt}$ . P is the population, is the intial value, is time, and is the growth constant. We can plug in the values we know at time t=0 and solve for .

$50=Ae^{0k}=A$

Now that we have solved for we can solve for at t=7

$200=50e^{7k}$

$e^{7k}=4$

$7k=\ln(4)$

$k=\frac{\ln(4)}{7}$

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Example Question #4 : How To Find Constant Of Proportionality Of Rate

A population of mice has 200 mice. After 6 weeks, there are 1600 mice in the population. What is the constant of growth?

Possible Answers:

$\frac{\ln(8)}{6}$

$\ln(8)$

$\frac{\ln(6)}{6}$

$\frac{\ln(6)}{8}$

Correct answer:

$\frac{\ln(8)}{6}$

Explanation:

The equation for population growth is given by $P=Ae^{kt}$ . is the population, is the intial value, is time, and is the growth constant. We can plug in the values we know at time t=0 and solve for .

$200=Ae^{0k}=A$

Now that we have we can solve for at t=6 .

$1600=200e^{6k}$

$e^{6k}=8$

$6k=\ln(8)$

$k=\frac{\ln(8)}{6}$

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Example Question #4 : Constant Of Proportionality

Find the direct constant of proportionality of y=3sec(x) from x=0 to $x=\frac{\pi}{3}$ .

Possible Answers:

$k=\frac{9}{\pi}$

$k=-\frac{\pi}{9}$

$k=\frac{\pi}{9}$

is undefined

Correct answer:

$k=\frac{9}{\pi}$

Explanation:

Direct constant of proportionality is given by

$k=\frac{\Delta y}{\Delta x}$ , where $\Delta y$ is the change in the position and $\Delta x$ is the change in the position.

Since y=3sec(x) , and we're going from x=0 to $x=\frac{\pi}{3}$

$\Delta y=3sec(\frac{\pi}{3})-3sec(0)=6-3=3$

$\Delta x=\frac{\pi}{3}-0=\frac{\pi}{3}$

$k=\frac{3}{\frac{\pi}{3}}=\frac{9}{\pi}$

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Example Question #2 : Constant Of Proportionality

The rate of decrease of the dwindling wolf population of Zion National Park is proportional to the population. The population decreased by 7 percent between 2009 and 2011. What is the constant of proportionality?

Possible Answers:

0.130

-0.011

-0.009

-0.044

-0.036

Correct answer:

-0.036

Explanation:

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population decreased by 7 percent between 2009 and 2011, we can solve for this constant of proportionality:

$(1-0.07)y_0=y_0e^{k(2011-2009)}$

$0.93=e^{2k}$

2k=ln(0.93)

$k=\frac{ln(0.93)}{2}=-0.036$

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Example Question #2 : Constant Of Proportionality

The rate of growth of the Martian Transgalactic Constituency is proportional to the population. The population increased by 23 percent between 2530 and 2534 AD. What is the constant of proportionality?

Possible Answers:

0.07

0.05

1.04

0.12

-0.37

Correct answer:

0.05

Explanation:

We're told that the rate of growth of the population is proportional to the population itself, meaning that this problem deals with exponential growth/decay. The population can be modeled thusly:

$p(t)=p_0e^{kt}$

Where p_0 is an initial population value, and is the constant of proportionality.

Since the population increased by 23 percent between 2530 and 2534 AD, we can solve for this constant of proportionality:

$(1+0.23)y_0=y_0e^{k(2534-2530)}$

$1.23=e^{4k}$

4k=ln(1.23)

$k=\frac{ln(1.23)}{4}=0.05$

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Calculus 1 : How to find constant of proportionality of rate

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #1 : How To Find Constant Of Proportionality Of Rate

Example Question #1 : Constant Of Proportionality

Example Question #2 : Constant Of Proportionality

Example Question #1 : Constant Of Proportionality

Example Question #2 : How To Find Constant Of Proportionality Of Rate

Example Question #1 : Constant Of Proportionality

Example Question #4 : How To Find Constant Of Proportionality Of Rate

Example Question #4 : Constant Of Proportionality

Example Question #2 : Constant Of Proportionality

Example Question #2 : Constant Of Proportionality

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