A Family Needs To Buy One Shovel And Between One And Eight Plants, Inclusive, For Their New Garden. The Cost Of The Shovel Is $s$ Dollars, And The Cost Of One Plant Is $p$ Dollars. Which Of The Following Functions Give The Possible

Introduction

When it comes to gardening, having the right tools and plants is essential for a successful and thriving garden. In this scenario, a family needs to purchase one shovel and between one and eight plants, inclusive, for their new garden. The cost of the shovel is $s$ dollars, and the cost of one plant is $p$ dollars. In this article, we will explore the possible cost functions that represent the total cost of the family's garden shopping.

Understanding the Cost Functions

To determine the possible cost functions, we need to consider the different scenarios that the family may encounter. The family must purchase one shovel, which costs $s$ dollars. In addition to the shovel, they may also purchase between one and eight plants, inclusive. Each plant costs $p$ dollars.

Scenario 1: Purchasing One Plant

If the family decides to purchase only one plant, the total cost will be the cost of the shovel plus the cost of one plant. This can be represented by the function:

$$C(1) = s + p$$

where $C(1)$ is the total cost when purchasing one plant.

Scenario 2: Purchasing Two Plants

If the family decides to purchase two plants, the total cost will be the cost of the shovel plus the cost of two plants. This can be represented by the function:

$$C(2) = s + 2p$$

where $C(2)$ is the total cost when purchasing two plants.

Scenario 3: Purchasing Three Plants

If the family decides to purchase three plants, the total cost will be the cost of the shovel plus the cost of three plants. This can be represented by the function:

$$C(3) = s + 3p$$

where $C(3)$ is the total cost when purchasing three plants.

Scenario 4: Purchasing Four Plants

If the family decides to purchase four plants, the total cost will be the cost of the shovel plus the cost of four plants. This can be represented by the function:

$$C(4) = s + 4p$$

where $C(4)$ is the total cost when purchasing four plants.

Scenario 5: Purchasing Five Plants

If the family decides to purchase five plants, the total cost will be the cost of the shovel plus the cost of five plants. This can be represented by the function:

$$C(5) = s + 5p$$

where $C(5)$ is the total cost when purchasing five plants.

Scenario 6: Purchasing Six Plants

If the family decides to purchase six plants, the total cost will be the cost of the shovel plus the cost of six plants. This can be represented by the function:

$$C(6) = s + 6p$$

where $C(6)$ is the total cost when purchasing six plants.

Scenario 7: Purchasing Seven Plants

If the family decides to purchase seven plants, the total cost will be the cost of the shovel plus the cost of seven plants. This can be represented by the function:

$$C(7) = s + 7p$$

where $C(7)$ is the total cost when purchasing seven plants.

Scenario 8: Purchasing Eight Plants

If the family decides to purchase eight plants, the total cost will be the cost of the shovel plus the cost of eight plants. This can be represented by the function:

$$C(8) = s + 8p$$

where $C(8)$ is the total cost when purchasing eight plants.

Generalizing the Cost Functions

From the previous scenarios, we can see that the total cost of the family's garden shopping can be represented by the function:

$$C(n) = s + np$$

where $C(n)$ is the total cost when purchasing $n$ plants, and $n$ is the number of plants purchased.

Conclusion

In conclusion, the possible cost functions that represent the total cost of the family's garden shopping are:

• $C(1) = s + p$
• $C(2) = s + 2p$
• $C(3) = s + 3p$
• $C(4) = s + 4p$
• $C(5) = s + 5p$
• $C(6) = s + 6p$
• $C(7) = s + 7p$
• $C(8) = s + 8p$

These functions can be generalized to:

$$C(n) = s + np$$

where $C(n)$ is the total cost when purchasing $n$ plants, and $n$ is the number of plants purchased.

References

• [1] Algebra, 2nd Edition, Michael Artin, Prentice Hall, 2010.
• [2] Calculus, 3rd Edition, Michael Spivak, Publish or Perish, 2008.

Further Reading

For further reading on cost functions and algebra, we recommend the following resources:

• [1] Khan Academy, Algebra Course.
• [2] MIT OpenCourseWare, Calculus Course.

Note: The references and further reading section are for educational purposes only and are not intended to be a comprehensive list of resources.

Introduction

In our previous article, we explored the possible cost functions that represent the total cost of a family's garden shopping. The family needs to purchase one shovel and between one and eight plants, inclusive, for their new garden. The cost of the shovel is $s$ dollars, and the cost of one plant is $p$ dollars. In this article, we will answer some frequently asked questions about the cost functions and provide additional insights.

Q&A

Q: What is the total cost of purchasing one plant?

A: The total cost of purchasing one plant is $s + p$ dollars.

Q: What is the total cost of purchasing two plants?

A: The total cost of purchasing two plants is $s + 2p$ dollars.

Q: Can I use the cost function to calculate the total cost of purchasing any number of plants?

A: Yes, you can use the cost function $C(n) = s + np$ to calculate the total cost of purchasing any number of plants.

Q: What is the cost function for purchasing eight plants?

A: The cost function for purchasing eight plants is $C(8) = s + 8p$.

Q: How do I calculate the total cost of purchasing a different number of plants?

A: To calculate the total cost of purchasing a different number of plants, simply substitute the number of plants into the cost function $C(n) = s + np$.

Q: Can I use the cost function to compare the costs of different numbers of plants?

A: Yes, you can use the cost function to compare the costs of different numbers of plants. For example, you can calculate the total cost of purchasing 5 plants and compare it to the total cost of purchasing 6 plants.

Q: What is the relationship between the cost function and the number of plants?

A: The cost function $C(n) = s + np$ shows a linear relationship between the total cost and the number of plants.

Q: Can I use the cost function to make predictions about the total cost of purchasing a certain number of plants?

A: Yes, you can use the cost function to make predictions about the total cost of purchasing a certain number of plants. For example, if you know the cost of the shovel and the cost of one plant, you can use the cost function to predict the total cost of purchasing 10 plants.

Conclusion

In conclusion, the cost function $C(n) = s + np$ provides a simple and effective way to calculate the total cost of purchasing any number of plants. By understanding the cost function and its relationship to the number of plants, you can make informed decisions about your garden shopping and compare the costs of different numbers of plants.

Additional Insights

• The cost function can be used to compare the costs of different numbers of plants.
• The cost function shows a linear relationship between the total cost and the number of plants.
• The cost function can be used to make predictions about the total cost of purchasing a certain number of plants.

References

• [1] Algebra, 2nd Edition, Michael Artin, Prentice Hall, 2010.
• [2] Calculus, 3rd Edition, Michael Spivak, Publish or Perish, 2008.

Further Reading

For further reading on cost functions and algebra, we recommend the following resources:

• [1] Khan Academy, Algebra Course.
• [2] MIT OpenCourseWare, Calculus Course.

Note: The references and further reading section are for educational purposes only and are not intended to be a comprehensive list of resources.