Question

A plant can manufacture 50 golf clubs per day at a total daily cost of $5363 and 80 golf clubs per day for a total cost of $7613. (A) Assuming that daily cost and production are linearly related, find the total daily cost, C, of producing x golf clubs. (B) Graph the total daily cost for 0≤x≤200. (C) Interpret the slope and y intercept of the cost equation. C=___

Answer 1

The total daily cost of producing x golf clubs can be found using linear regression. The equation of the line representing the total daily cost is C = 7.5x + 4988.

Step-by-step explanation:

The total daily cost, C, of producing x golf clubs can be found using linear regression. We have two data points:

Point 1: (50, 5363) - where x = 50 and C = 5363

Point 2: (80, 7613) - where x = 80 and C = 7613

Using the point-slope form of a linear equation, we can find the equation of the line that represents the relationship between the total daily cost and the number of golf clubs produced. Let's start by finding the slope:

slope = (change in C) / (change in x) = (7613 - 5363) / (80 - 50) = 225 / 30 = 7.5

Next, we can use one of the given data points to find the y-intercept:

5363 = 7.5 * 50 + b ⇒ b = 5363 - 7.5 * 50 = 5363 - 375 = 4988

So, the equation of the line is C = 7.5x + 4988. This equation represents the total daily cost, C, of producing x golf clubs.

Answer 2

The total daily cost, C, of producing x golf clubs can be found using the equation C = 7.5x + 4988. The graph of the total daily cost for 0≤x≤200 is a straight line with slope 7.5 and y-intercept 4988. The slope represents the additional cost incurred for each additional golf club produced, and the y-intercept represents the fixed cost.

Step-by-step explanation:

(A) To find the total daily cost, C, of producing x golf clubs, we can use the equation of a line. Let's assume the equation is given by C = mx + b, where m is the slope and b is the y-intercept. We can find the values of m and b using the given information:

1. When the plant manufactures 50 golf clubs per day, the total daily cost is $5363. This gives us the point (50, 5363).

2. When the plant manufactures 80 golf clubs per day, the total daily cost is $7613. This gives us the point (80, 7613).

Now we can use these two points to find the values of m and b. First, let's find the slope:

m = (y2 - y1) / (x2 - x1) = (7613 - 5363) / (80 - 50) = 225 / 30 = 7.5

Next, let's substitute the values of one of the points into the equation to find b:

5363 = 7.5 * 50 + b

b = 5363 - 7.5 * 50 = 5363 - 375 = 4988

So the equation for the total daily cost, C, of producing x golf clubs is C = 7.5x + 4988.

(B) To graph the total daily cost for 0≤x≤200, we can substitute different values of x into the equation C = 7.5x + 4988 and plot the corresponding values of C. The graph will be a straight line with slope 7.5 and y-intercept 4988.

(C) The slope of the cost equation, 7.5, represents the additional cost incurred for each additional golf club produced. The y-intercept of the cost equation, 4988, represents the fixed cost or the cost of production even when no golf clubs are produced.