Sarah Wants To Print Copies Of Her Artwork. At The Local Print Shop, It Costs Her $1$ To Make 5 Copies And $5$ To Make 25 Copies. What Is The Constant Of Proportionality In This Direct Variation?A. $\frac{1}{5}$ B. 5 C. 1

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What is Direct Variation?

Direct variation is a type of mathematical relationship where one quantity is directly proportional to another quantity. In other words, as one quantity increases, the other quantity also increases at a constant rate. This relationship can be represented by the equation y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of proportionality.

The Problem: Printing Copies of Artwork

Sarah wants to print copies of her artwork at a local print shop. The cost of printing is as follows:

  • It costs her $1 to make 5 copies.
  • It costs her $5 to make 25 copies.

We need to find the constant of proportionality in this direct variation.

Step 1: Identify the Given Information

We are given two sets of information:

  • 5 copies cost $1
  • 25 copies cost $5

Step 2: Determine the Constant of Proportionality

To find the constant of proportionality, we need to determine the rate at which the cost increases as the number of copies increases. We can do this by dividing the cost by the number of copies.

For the first set of information, we have:

Cost = $1 Number of copies = 5

Rate = Cost / Number of copies = $1 / 5 = $0.20 per copy

For the second set of information, we have:

Cost = $5 Number of copies = 25

Rate = Cost / Number of copies = $5 / 25 = $0.20 per copy

As we can see, the rate is the same in both cases, which means that the cost increases at a constant rate. This is a direct variation.

Step 3: Find the Constant of Proportionality

Now that we have determined the rate, we can find the constant of proportionality (k) by multiplying the rate by the number of copies.

For the first set of information, we have:

Rate = $0.20 per copy Number of copies = 5

k = Rate x Number of copies = $0.20 x 5 = $1

For the second set of information, we have:

Rate = $0.20 per copy Number of copies = 25

k = Rate x Number of copies = $0.20 x 25 = $5

As we can see, the constant of proportionality is the same in both cases, which means that the relationship is a direct variation.

Conclusion

In this problem, we have found the constant of proportionality in the direct variation between the cost of printing and the number of copies. The constant of proportionality is $1, which means that for every 5 copies printed, the cost increases by $1.

Answer

The constant of proportionality in this direct variation is $1.

Discussion

This problem illustrates the concept of direct variation and how to find the constant of proportionality. Direct variation is a fundamental concept in mathematics and is used to model many real-world relationships. Understanding direct variation and how to find the constant of proportionality is essential for solving problems in mathematics and science.

Real-World Applications

Direct variation has many real-world applications, including:

  • Physics: The relationship between distance, speed, and time is a direct variation.
  • Economics: The relationship between supply and demand is a direct variation.
  • Engineering: The relationship between stress and strain is a direct variation.

Conclusion

Q: What is direct variation?

A: Direct variation is a type of mathematical relationship where one quantity is directly proportional to another quantity. In other words, as one quantity increases, the other quantity also increases at a constant rate.

Q: What is the equation for direct variation?

A: The equation for direct variation is y = kx, where y is the dependent variable, x is the independent variable, and k is the constant of proportionality.

Q: What is the constant of proportionality?

A: The constant of proportionality (k) is a value that represents the rate at which one quantity increases as the other quantity increases. It is a key concept in direct variation.

Q: How do I find the constant of proportionality?

A: To find the constant of proportionality, you need to determine the rate at which one quantity increases as the other quantity increases. You can do this by dividing the dependent variable by the independent variable.

Q: What is an example of direct variation in real life?

A: An example of direct variation in real life is the relationship between the distance traveled and the time taken to travel that distance. As the distance traveled increases, the time taken to travel that distance also increases at a constant rate.

Q: Can you give me an example of a direct variation problem?

A: Here's an example:

A car travels 250 miles in 5 hours. How far will it travel in 10 hours?

To solve this problem, you need to find the constant of proportionality (k) by dividing the distance traveled by the time taken:

k = 250 miles / 5 hours = 50 miles per hour

Now that you have the constant of proportionality, you can use it to find the distance traveled in 10 hours:

Distance = k x time = 50 miles per hour x 10 hours = 500 miles

Q: What is the difference between direct variation and inverse variation?

A: Direct variation and inverse variation are two types of mathematical relationships. In direct variation, one quantity increases as the other quantity increases at a constant rate. In inverse variation, one quantity decreases as the other quantity increases at a constant rate.

Q: Can you give me an example of an inverse variation problem?

A: Here's an example:

A light bulb uses 60 watts of power when it is 6 feet away from a heat source. How much power will it use when it is 12 feet away?

To solve this problem, you need to find the constant of proportionality (k) by dividing the power used by the distance:

k = 60 watts / 6 feet = 10 watts per foot

Now that you have the constant of proportionality, you can use it to find the power used when the light bulb is 12 feet away:

Power = k x distance = 10 watts per foot x 12 feet = 120 watts

Q: How do I graph a direct variation?

A: To graph a direct variation, you need to plot the dependent variable (y) against the independent variable (x). The graph will be a straight line with a positive slope.

Q: Can you give me an example of a direct variation graph?

A: Here's an example:

Suppose we have a direct variation between the distance traveled and the time taken to travel that distance. The equation for this variation is y = 50x, where y is the distance traveled and x is the time taken.

To graph this variation, we need to plot the distance traveled (y) against the time taken (x). The graph will be a straight line with a positive slope.

Conclusion

In conclusion, direct variation is a fundamental concept in mathematics that describes a relationship between two quantities where one quantity is directly proportional to another quantity. The constant of proportionality is a key concept in direct variation that represents the rate at which one quantity increases as the other quantity increases. Understanding direct variation and how to find the constant of proportionality is essential for solving problems in mathematics and science.