Grant Is Using A New Weed Killer To Eliminate The Weeds In His Garden. The Number Of Weeds Changes According To The Function F ( X ) = 100 ⋅ ( 1 2 ) X F(x)=100 \cdot\left(\frac{1}{2}\right)^x F ( X ) = 100 ⋅ ( 2 1 ) X , Where X X X Represents The Number Of Weeks Since He Began Using The

Mar 9, 2025 by ADMIN 290 views

**The Exponential Decline of Weeds in Grant's Garden**

Understanding the Problem

Grant is using a new weed killer to eliminate the weeds in his garden. The number of weeds changes according to the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ , where $x$ represents the number of weeks since he began using the weed killer. This function represents an exponential decline in the number of weeds, with the rate of decline being 50% each week.

The Exponential Function

The function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ is an example of an exponential function. Exponential functions have the form $f(x)=a \cdot b^x$ , where $a$ is the initial value and $b$ is the growth or decay factor. In this case, the initial value is 100, and the decay factor is $\frac{1}{2}$ .

How the Function Works

To understand how the function works, let's break it down step by step. The function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ can be rewritten as $f(x)=100 \cdot \frac{1}{2^x}$ . This means that the number of weeds is equal to 100 divided by $2^x$ .

Calculating the Number of Weeds

To calculate the number of weeds at any given week, we can plug in the value of $x$ into the function. For example, if Grant has been using the weed killer for 1 week, the number of weeds would be:

$f(1)=100 \cdot \frac{1}{2^1}=50$

If Grant has been using the weed killer for 2 weeks, the number of weeds would be:

$f(2)=100 \cdot \frac{1}{2^2}=25$

And if Grant has been using the weed killer for 3 weeks, the number of weeds would be:

$f(3)=100 \cdot \frac{1}{2^3}=12.5$

The Rate of Decline

The rate of decline in the number of weeds is 50% each week. This means that if Grant has 100 weeds one week, he will have 50 weeds the next week, and 25 weeds the week after that.

Graphing the Function

The graph of the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ is a curve that starts at 100 and declines exponentially. The graph can be plotted using a graphing calculator or a computer program.

Real-World Applications

The function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ has many real-world applications. For example, it can be used to model the decline of a population over time, or the decay of a radioactive substance.

Conclusion

In conclusion, the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ is an example of an exponential function that models the decline of weeds in Grant's garden. The function has a rate of decline of 50% each week, and can be used to calculate the number of weeds at any given week. The graph of the function is a curve that starts at 100 and declines exponentially.

Mathematical Derivations

To derive the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ , we can start with the equation $f(x)=100 \cdot \frac{1}{2^x}$ . We can rewrite this equation as $f(x)=100 \cdot \frac{1}{2^x} = 100 \cdot \frac{1}{2^x} \cdot \frac{2^x}{2^x} = 100 \cdot \frac{2^x}{2^{x+1}} = 100 \cdot \frac{1}{2}$ .

Solving for x

To solve for $x$ , we can set the equation $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ equal to a given value, and then solve for $x$ . For example, if we want to find the value of $x$ when $f(x)=25$ , we can set up the equation $25=100 \cdot\left(\frac{1}{2}\right)^x$ and then solve for $x$ .

Using the Function to Model Real-World Scenarios

The function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ can be used to model many real-world scenarios. For example, it can be used to model the decline of a population over time, or the decay of a radioactive substance.

Case Study: Decline of a Population

Suppose we want to model the decline of a population over time. We can use the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ to model the decline of the population. For example, if the initial population is 100, and the population declines by 50% each week, we can use the function to calculate the population at any given week.

Case Study: Decay of a Radioactive Substance

Suppose we want to model the decay of a radioactive substance. We can use the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ to model the decay of the substance. For example, if the initial amount of the substance is 100, and the substance decays by 50% each week, we can use the function to calculate the amount of the substance at any given week.

Conclusion

Q: What is the function that models the decline of weeds in Grant's garden?

A: The function that models the decline of weeds in Grant's garden is $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ , where $x$ represents the number of weeks since Grant began using the weed killer.

Q: What is the rate of decline of the weeds?

A: The rate of decline of the weeds is 50% each week. This means that if Grant has 100 weeds one week, he will have 50 weeds the next week, and 25 weeds the week after that.

Q: How can I use the function to calculate the number of weeds at any given week?

A: To calculate the number of weeds at any given week, you can plug in the value of $x$ into the function. For example, if Grant has been using the weed killer for 1 week, the number of weeds would be $f(1)=100 \cdot \frac{1}{2^1}=50$ .

Q: What is the graph of the function like?

A: The graph of the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ is a curve that starts at 100 and declines exponentially. The graph can be plotted using a graphing calculator or a computer program.

Q: Can the function be used to model real-world scenarios?

A: Yes, the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ can be used to model many real-world scenarios, including the decline of a population over time, or the decay of a radioactive substance.

Q: How can I derive the function?

A: To derive the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ , you can start with the equation $f(x)=100 \cdot \frac{1}{2^x}$ . You can then rewrite this equation as $f(x)=100 \cdot \frac{1}{2^x} \cdot \frac{2^x}{2^x} = 100 \cdot \frac{2^x}{2^{x+1}} = 100 \cdot \frac{1}{2}$ .

Q: Can I use the function to solve for x?

A: Yes, you can use the function to solve for $x$ . For example, if you want to find the value of $x$ when $f(x)=25$ , you can set up the equation $25=100 \cdot\left(\frac{1}{2}\right)^x$ and then solve for $x$ .

Q: What are some real-world applications of the function?

A: Some real-world applications of the function $f(x)=100 \cdot\left(\frac{1}{2}\right)^x$ include modeling the decline of a population over time, or the decay of a radioactive substance. The function can also be used to model the decline of weeds in a garden, as in the case of Grant's garden.

Q: How can I use the function to model the decline of a population?

A: To use the function to model the decline of a population, you can set the initial population equal to 100, and then use the function to calculate the population at any given week. For example, if the initial population is 100, and the population declines by 50% each week, you can use the function to calculate the population at any given week.

Q: How can I use the function to model the decay of a radioactive substance?

A: To use the function to model the decay of a radioactive substance, you can set the initial amount of the substance equal to 100, and then use the function to calculate the amount of the substance at any given week. For example, if the initial amount of the substance is 100, and the substance decays by 50% each week, you can use the function to calculate the amount of the substance at any given week.