Problem 44 - Algebra 1

Question

Chai has a small business making decorated hats. She calculates her monthly cost \(y\) of producing \(x\) hats using the function \(y=1.9 x+350 .\) In January, her cost was \(\$ 458.30 .\) How many hats did she make that month? Solve algebraically and graphically.

Step-by-Step Solution

Answer

Chai made approximately 57 hats in January. The graphical representation will show the corresponding point on the line \(y = 1.9x + 350\) drawn against cost and number of hats.

Step 1: Substitute the given value of y

To find the number of hats Chai made in January, we need to substitute the given value of \(y\), which is the cost for January into the equation. Thus we have: \(458.30 = 1.9x + 350 \)

Step 2: Solve for x

In order to isolate \(x\) (the number of hats), we subtract 350 from both sides of the equation: \(458.30 - 350 = 1.9x \). From this, we get \(108.3 = 1.9x\). Then, we divide both sides of the resulting equation by 1.9 to find the value of \(x\). i.e., \(x = 108.3 / 1.9 \).

Step 3: Calculate the numerical value of x

After calculation, we find \(x \approx 57\). Thus, Chai made approximately 57 hats in January.

Step 4: Graphical representation

Now, we create a graph with cost on the y-axis and number of hats on the x-axis. We plot the line using the equation \(y = 1.9x + 350 \), and we confirm the point (57, 458.30) lies on this line.

Key Concepts

Algebraic Solving

To solve a linear equation like the one in this exercise, we use algebraic techniques to find the unknown variable, which in this case is the number of hats, represented by \( x \). The process begins by substituting a known value, \( y = 458.30 \), into the equation \( y = 1.9x + 350 \). This gives us the equation \( 458.30 = 1.9x + 350 \).
To isolate \( x \), subtract 350 from both sides: \( 458.30 - 350 = 1.9x \), leading to \( 108.3 = 1.9x \).
The final step requires dividing both sides by 1.9 to solve for \( x \), which yields \( x = \frac{108.3}{1.9} \). Calculating this results in \( x = 57 \), meaning Chai made approximately 57 hats in that month.
These steps demonstrate the systematic approach of algebraic solving, ensuring accuracy when finding unknown values.

Graphical Representation

Visualizing linear equations helps to understand the relationship between variables. In this exercise, we graph the function \( y = 1.9x + 350 \) to represent the costs related to the number of hats.

• The x-axis represents the number of hats \( x \).
• The y-axis represents the cost \( y \).

We plot the equation on a graph to see the linear relationship. The line will slope upwards, indicating that as the number of hats increases, the cost also increases.
By adding the calculated point (57, 458.30) to the graph, we check if it lies on the line. Points on the line satisfy the equation, confirming Chai’s production number is accurate.
Graphical representation thus validates algebraic solutions and offers a visual confirmation of the relationship between variables.

Function Application

A key concept here is the application of a function to model real-world scenarios. In this problem, the function \( y = 1.9x + 350 \) describes how production costs change with the quantity of hats made.
Functions serve as mathematical models that predict outcomes based on varying inputs.

• \( 1.9x \) shows the variable cost depending on the number of hats.
• The constant \( 350 \) represents the fixed cost, which does not change with production.

By inputting different values of \( x \), Chai can estimate her costs for any given number of hats.
This function application not only helps in predicting costs but also aids in planning and budgeting business activities efficiently. Understanding these functions is essential for making informed decisions in both personal and professional settings.