Your weekly cost (in dollars) to manufacture x cars and y...

Your weekly cost (in dollars) to manufacture $x$ cars and $y$ trucks is $$ C(x, y)=240,000+6,000 x+4,000 y $$ a. What is the marginal cost of a car? Of a truck? HINT [See Example 1.] b. Describe the graph of the cost function C. HINT [See Example 7.] c. Describe the slice $x=10$. What cost function does this slice describe? d. Describe the level curve $z=480,000$. What does this curve tell you about costs?

Your weekly cost (in dollars) to manufacture $x$ cars and $y$ trucks is
$$
C(x, y)=240,000+6,000 x+4,000 y
$$
a. What is the marginal cost of a car? Of a truck? HINT [See Example 1.]
b. Describe the graph of the cost function C. HINT [See Example 7.]
c. Describe the slice $x=10$. What cost function does this slice describe?
d. Describe the level curve $z=480,000$. What does this curve tell you about costs?

Key Concepts

Marginal Cost

Marginal cost represents the additional cost incurred by producing one more unit of a product. In the context of cost functions, it is defined by the partial derivative with respect to the product quantity. This concept helps in determining how production increases affect the overall cost, providing essential information for decision-making in production and pricing.

Cost Function

A cost function is a mathematical formula that models the total production cost as a function of quantities produced. It typically includes fixed costs, which do not change with production levels, and variable costs, which change in proportion to the number of units produced. This function provides insight into how costs behave with changes in production volume.

Graph of a Function

The graph of a cost function often represents a plane (if the function is linear with two variables) in a multidimensional coordinate system, where each axis corresponds to a product type or cost component. Analyzing the graph can reveal information about the behavior of the function, such as intercepts, slopes (which represent marginal costs), and overall trends in cost as production levels vary.

Function Slice

A slice of a multivariable function is obtained by holding one or more variables constant, thereby reducing the function to one of fewer variables. This technique allows for the analysis of the function along a specific dimension. In a cost function context, slicing the function by fixing the quantity of one product enables a clearer understanding of how the cost behaves with respect to changes in the production level of the other product.

Level Curves

Level curves, or contour lines, are curves on a graph where the function maintains a constant value. In the context of a cost function, each level curve represents all combinations of product quantities that yield the same total cost. Analyzing level curves aids in visualizing trade-offs between products and understanding how cost remains constant despite varying production levels.

Transcript

00:01 In this problem of functions of several variable, we have given that your weekly cost in dollars, so in dollars, to manufacture x cars and y trucks.

00:11 So, x is representing cars and y is representing trucks.

00:18 Is given why, say, c, this is a cost function, c of xy is equals to 240 ,000 plus 6 ,000 multiplied with x plus 4 ,000 multiplied with y so this is 4 ,000 so this is 4 ,000 dollars multiplied with y now it is asked that what is the marginal cost of a car and of a truck so x stand for car so marginal cost would be 6 ,000 for car so 6 ,000 dollar for car and 4 ,000 caller and 4 ,000 caller dollar actually so 4 ,000 dollars for trucks say for truck and now in part b part b say that describe the graph of the cost function c so we have to describe the graph so this is z is equals to so this can be written as z is equals to 240 ,000 plus 6 ,000 x plus 4 ,000 y so the graph would be a place with intercept say for finding the x intercept say x intercept putting x intercept because this is a linear function so x y and z involved so x intercept so for this we have to put y is equal to 0 and z is equal to 0 so putting y is equal to 0 and z is equal so x would be 240 ,000 divided with 6 ,000 so x is equal this would be minus 40 so x is equal to minus 40 and y intercept so y intercept so y intercept would be now putting x is zero zero x is zero and z is zero so x is equal to zero and z is also equals to zero so this turn would be zero and this turn would be zero so we get y is equal to minus 240 000 divided with 4 000 is equal to here minus 60 so y is equals to minus 60 so this is the y intercept and z interstate and so z intercept.

02:42 So for this we have to put x is equals to 0 and y is also equals to 0 so we get z is equals to 240 ,000 so we had 240 ,000 so we have the answer that is the graph is a plane so graph is a plane why it is plain because we have the linear function so graph is a plane and x is a plane intercept with say here with x intercept we can say x intercept is equals to say minus 40 y intercept say here y intercept so y intercept would be minus 60 and z intercept is say here z intercept is equal to say this is 243 3 so this is 240 ,000.

03:47 So this is the solution for part b, now part c.

03:49 In part c, we have to describe the slice at x is equals to 10...

Please give Ace some feedback