Let's tackle this problem step by step!

Part (A): Finding the cost equation

The problem states that the cost $C$ is linearly related to the number of golf clubs produced, so the general form of the equation for total daily cost $C$ in terms of the number of golf clubs produced $x$ is:

$C = mx + b$

Where:

• $m$ is the slope of the line (the cost per additional golf club produced)
• $b$ is the y-intercept (the fixed cost, i.e., the cost when 0 golf clubs are produced)

From the given data:

• When 50 golf clubs are produced, the cost is $4629: $(50, 4629)$
• When 80 golf clubs are produced, the cost is $6429: $(80, 6429)$

Using these two points, we can find the slope $m$:

$m = \frac{\text{Change in cost}}{\text{Change in quantity}} = \frac{6429 - 4629}{80 - 50} = \frac{1800}{30} = 60$

So, the slope of the line is 60, meaning the cost of producing each additional golf club is $60.

Next, we can use one of the points (let's use $(50, 4629)$) to find the y-intercept $b$:

$4629 = 60 \times 50 + b$

$4629 = 3000 + b$

$b = 4629 - 3000 = 1629$

Thus, the cost equation is:

$C = 60x + 1629$

Part (B): Graphing the total daily cost

To graph the total daily cost for $0 \leq x \leq 200$, we know:

• The equation of the line is $C = 60x + 1629$
• The slope $m = 60$, so for each increase in $x$, the cost increases by 60.
• The y-intercept $b = 1629$, so when $x = 0$, the cost is 1629.

At $x = 0$, $C = 1629$. At $x = 200$, $C = 60 \times 200 + 1629 = 12000 + 1629 = 13629$.

The line will go through the points $(0, 1629)$ and $(200, 13629)$, with a steady upward slope.

To graph this, plot these two points on a coordinate plane and draw a straight line through them. The line will be a straight diagonal line going upwards from left to right.

Part (C): Interpreting the slope and y-intercept

• The slope of the line is 60. This means that for each additional golf club produced, the total cost increases by $60.
• The y-intercept is 1629. This is the fixed cost, which represents the cost of producing 0 golf clubs. This is likely the base operating cost of the plant, which does not change regardless of production level.

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Summary

• (A) The total daily cost equation is:
  $C = 60x + 1629$.

• (B) The graph will be a straight line, with the cost starting at 1629 when $x = 0$, and increasing by 60 units for each additional golf club produced.

• (C) The slope $60$ represents the cost per golf club, and the y-intercept $1629$ is the fixed cost when no golf clubs are produced.

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Would you like further details or have any additional questions?

Here are 5 questions you might explore next:

1. How does changing the slope affect the total daily cost equation?
2. What happens to the fixed cost if the plant has no operational overhead?
3. How would the graph change if the production cost per golf club was $80 instead of $60?
4. How could you calculate the break-even point where costs equal revenues if you know the price per golf club?
5. What is the impact on the cost equation if there are fixed costs that change over time?

Tip: When interpreting linear cost equations, remember that the slope tells you the marginal cost (the cost for each additional unit), while the y-intercept shows the starting cost, even if no units are produced.