Problem 24 - Algebra A Combined Function

Question

Because it is more efficient to produce larger numbers of items, the cost of producing a certain computer DVD is inversely proportional to the number produced. If 4000 can be produced at a cost of \(\$ 1.20\) each, find the cost per DVD when 6000 are produced.

Step-by-Step Solution

Answer

The cost per DVD is \( \$ 0.80 \) when 6000 DVDs are produced.

Step 1: Understand the problem

The cost of producing DVDs is inversely proportional to the number of DVDs produced. This means as more DVDs are produced, the cost per DVD decreases. We are given the cost for producing 4000 DVDs and need to find the cost for 6000 DVDs.

Step 2: Set up the inverse proportionality relationship

The relationship can be expressed as \( C = \frac{k}{n} \), where \( C \) is the cost per DVD, \( n \) is the number of DVDs produced, and \( k \) is a constant. We know \( C = 1.20 \) when \( n = 4000 \). Substitute these values into the equation to find \( k \).

Step 3: Calculate the constant of proportionality

Using the formula \( C = \frac{k}{n} \), substitute the known values: \( 1.20 = \frac{k}{4000} \). Solve for \( k \) by multiplying both sides by 4000: \( k = 1.20 \times 4000 = 4800 \).

Step 4: Use the constant to find the new cost per DVD

Now that \( k = 4800 \), use the formula again for 6000 DVDs: \( C = \frac{k}{6000} = \frac{4800}{6000} \). Simplify this to find the cost per DVD.

Step 5: Simplify the expression

Calculate \( \frac{4800}{6000} \) which simplifies to \( 0.80 \). Thus, the cost per DVD when 6000 DVDs are produced is \( \$ 0.80 \).

Key Concepts

Cost Calculation

When dealing with inverse proportionality, calculating the cost becomes an interesting task.
In this context, we consider how the cost to produce each DVD changes when the total number of DVDs produced increases. Inverse proportionality means that as more items are produced, the cost per item decreases.
This is very common in large-scale production. For this DVD problem, let's break it down:

• When you produce more DVDs, the cost per DVD decreases. This happens because certain costs are "spread out" over all the DVDs being created.
• The goal is to figure out the new cost when we change the number of DVDs produced.

To do this, we first understand that the cost per DVD for 4000 units is \( \$1.20 \). By leveraging a formula derived from the concept of inverse proportionality, we can adjust this cost when the production number changes, ensuring efficient cost calculation.

Constant of Proportionality

In solving problems involving inverse proportionality, a key step is finding the "constant of proportionality."
This constant is crucial, as it allows us to apply a standard proportional change regardless of the number of items.
Here's how we identify and use this constant:* To find this constant, we use the known cost and quantity. The formula for inverse proportionality is \( C = \frac{k}{n} \).*** We plug in the values, where \( C = 1.20 \) and \( n = 4000 \), to find \( k \).* Solving the equation \( 1.20 = \frac{k}{4000} \), we multiply both sides by \( 4000 \), resulting in \( k = 4800 \).Understanding the constant of proportionality helps us easily compute the cost per DVD for any number of DVDs produced, using the same relationship.

Simplifying Expressions

Mathematical expressions are essential for clear and concise computation. Simplifying an expression makes it easier to understand and work with, which is particularly useful in proportional relationships.
In our exercise:* After finding the constant of proportionality (\( k = 4800 \)), we calculate the cost when producing 6000 DVDs using the formula \( C = \frac{k}{n} \).* Plugging in the values: \( C = \frac{4800}{6000} \).* Simplifying the fraction \( \frac{4800}{6000} \) is key. By dividing both the numerator and the denominator by 600, we simplify this to 0.8, which means the cost per DVD is \( \$ 0.80 \).Simplifying expressions effectively leads us to cleaner and more understandable results, easing the computational process.