Exponential Growth and Decay Worksheet

We have created logarithmic functions and exponential from a mathematical point of view. This includes defining functions and their derivatives, graphs, integrals, properties, etc. We also consider Exponential Growth and Decay in other fields. By the end of this article, you’ll easily understand the Exponential Growth and Decay worksheet with many examples. This also has explained multiple scientific phenomena. There are some of the applications of growth and decay in the Exponential Growth and Decay worksheet, which are compound interest, radioactive decay, population growth, etc.

To find the data of the ancient objects, archaeologists at archaeological sites use the radioactive decay method.

Exponential Functions

Let us look into the functions given below in the Exponential Growth and Decay Worksheet;

y=x2, here we have the fixed exponent, and we can see the base is the variable. So we now know it is a quadratic function.

y=2x, here the exponent can be seen as a variable, and we can see the fixed base, so an exponential function.

We can define an exponential function as the function that has the general formulae in the form of y = abx

Here,

a ≠ 0 and b>0 and is real also b ≠ 1.

b is a constant in an exponential function, and the x exponent is the independent variable.

Here we can see that only the set of real numbers is the domain.

Exponential functions are of two types:

• Exponential growth
• Exponential decay

If f (x) = bx when b > 1, function denotes the exponential growth.

If f (x) = bx when 0 < b < 1, function denotes exponential decay.

Exponential Growth and Decay worksheet

Growth

The idea of growth is amazing. We can understand this by the doubling method. Like there is something that keeps on growing as compared to its original value. Let’s take an example if the population of dogs doubles every month. Then will have 2 dogs in the first month. 4 dogs in the second month. Eight dogs in the third month. Sixteen dogs in the fourth month, and this keeps increasing as the month passes by.

dogs= ex

where ‘e’ is Euler’s number. This denotes exponential growth.

Decay

We can understand this as the opposite of growth. We have a formula;

y(t) = a × ekt

Here,

y(t) = value at time “t”

a = value at the start

k = rate of growth (when >0) or decay (when <0)

t = time

Sometimes the things get smaller, or we can say things decay this exponential decay easily. We can understand this by the example of atmospheric pressure.

Atmospheric pressure is the pressure of the air that is around you. This keeps on decreasing as we go higher, that is 12% for every 1000m. this is an exponential decay. By using the decay formulae, you can easily find out the atm pressure.

Half-life

This term is for radioactive decay and also has many different applications. The time taken for a value to halve with its exponential decay is the half-life. The formulae used to calculate the half-life is

y(t) = a × ekt

here,

a is the starting point.

t is the time taken.

y(t) is the reduction of half-life.

Exponential Word Problem

Here we’ll know about the exponential function word problem worksheet with answers.

• A group of ants is growing exponentially. At time t=0, it has 10 ants in it, and at time t=4, it has 2000. At what time will it have 100,000 ants?

Solution

Although it’s not demanded in Exponential Word Problem, we have to find the general formula for the number f(t) of ants at time t, let this expression is equal to 100,000, and solve for t.

Now, we will take a shortcut here since we know that c=f(0)

And we know f (0) =10.

And use the formula for k:

k=lnf(t1) −lnf(t2)/t1−t2=ln10−ln2,000/0−4=ln10/2,000/−4=ln200/4

Thus, we have

f(t)=10⋅eln200/4t=10⋅200t/4

as the general formula.

Now we solve

100,000=10⋅eln200/4t

for t: divide both sides by 10 and take logarithms to get

ln10,000=ln200/4t

Thus,

t=4ln10,000/ln200≈6.953407835.

Conclusion

The most common way by which we can see the growth or decay of this is exponential. In this article, we have discussed the exponential growth and decay worksheet by mathematical models, be it biological, social, chemical, or physical. We also consider other equations and models.

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