Question

1. production cost a small - appliance manufacturer finds that if he produces x toaster ovens in a month, his production cost is given by the equation

$y = 6x + 3000$
(where y is measured in dollars).
(a) sketch a graph of this linear equation.
(b) what do the slope and y - intercept of the graph represent?

Explanation:

Part (a)

Step 1: Identify the form of the equation

The equation \( y = 6x + 3000 \) is in the slope - intercept form of a linear equation \( y=mx + b \), where \( m \) is the slope and \( b \) is the \( y \) - intercept. Here, \( m = 6 \) and \( b=3000 \).

Step 2: Find two points on the line

• When \( x = 0 \):

Substitute \( x = 0 \) into the equation \( y=6x + 3000 \). Then \( y=6(0)+3000=3000 \). So one point is \( (0,3000) \).

• When \( x = 500 \) (we can choose any non - zero value for \( x \), 500 is a convenient choice):

Substitute \( x = 500 \) into the equation \( y = 6x+3000 \). Then \( y=6\times500 + 3000=3000 + 3000=6000 \). So another point is \( (500,6000) \).

Step 3: Sketch the graph

• Draw a coordinate system where the \( x \) - axis represents the number of toaster ovens (\( x \)) and the \( y \) - axis represents the production cost in dollars (\( y \)).
• Plot the points \( (0,3000) \) and \( (500,6000) \).
• Draw a straight line passing through these two points. The line should have a positive slope (since \( m = 6>0 \)) and intersect the \( y \) - axis at \( (0,3000) \).

Part (b)

Step 1: Interpret the slope (\( m = 6 \))

The slope of a linear equation in the context of \( y \) (production cost) and \( x \) (number of toaster ovens) is given by \( m=\frac{\Delta y}{\Delta x} \). In the equation \( y = 6x+3000 \), the slope \( m = 6 \) means that for each additional toaster oven produced (each increase of 1 in the value of \( x \)), the production cost (\( y \)) increases by 6 dollars. In other words, the slope represents the variable cost per toaster oven, which is \$6 per toaster oven.

Step 2: Interpret the \( y \) - intercept (\( b = 3000 \))

The \( y \) - intercept occurs when \( x = 0 \) (when the number of toaster ovens produced is 0). Substituting \( x = 0 \) into the cost equation \( y=6x + 3000 \) gives \( y = 3000 \). This means that when no toaster ovens are produced (\( x = 0 \)), the production cost is \$3000. This represents the fixed cost of production, the cost that the manufacturer incurs even if no toaster ovens are made (for example, costs of equipment, rent, etc. that do not depend on the number of units produced).

Answer:

s:
(a) The graph is a straight line passing through \((0, 3000)\) and \((500, 6000)\) (or other suitable points) with \(x\) as the number of toaster ovens and \(y\) as cost. (Graph description as above)
(b) Slope: The variable cost per toaster oven (\$6 per toaster oven). LXI0 - intercept: The fixed production cost (\$3000 when \(x = 0\)).