Problem 2 - The Calculus with Analytic Geometry

Question

If \(C(x)\) dollars is the total cost of manufacturing \(x\) toys and \(C(x)=110+4 x+0.02 x^{2}\), find (a) the marginal cost function, (b) the marginal cost when \(x=10\), and (c) the cost of manufacturing the eleventh toy.

Step-by-Step Solution

Answer

(a) MC(x) = 4 + 0.04x. (b) 4.4. (c) 4.42.

Step 1 - Find the Marginal Cost Function

The marginal cost function is the derivative of the total cost function with respect to the number of toys, x. Start by differentiating the given cost function: \[ C(x) = 110 + 4x + 0.02x^2 \] So, the marginal cost function, \( MC(x) \), is: \[ MC(x) = \frac{d}{dx} (110 + 4x + 0.02x^2) \] Calculate the derivative term by term: \[ MC(x) = 0 + 4 + 0.04x \] Therefore, the marginal cost function is: \[ MC(x) = 4 + 0.04x \]

Step 2 - Calculate the Marginal Cost When x=10

Substitute x = 10 into the marginal cost function: \( MC(x) = 4 + 0.04x \). Thus, when \( x = 10 \): \[ MC(10) = 4 + 0.04(10) = 4 + 0.4 = 4.4 \] So, the marginal cost when 10 toys are manufactured is \( 4.4 \) dollars.

Step 3 - Calculate the Cost of Manufacturing the Eleventh Toy

To find the cost of manufacturing the eleventh toy, calculate the total cost for 11 toys and subtract the total cost of 10 toys: \[ C(11) = 110 + 4(11) + 0.02(11^2) \] Calculate: \[ C(11) = 110 + 44 + 0.02(121) = 110 + 44 + 2.42 = 156.42 \] Next, find \( C(10) \): \[ C(10) = 110 + 4(10) + 0.02(100) \] Calculate: \[ C(10) = 110 + 40 + 2 = 152 \] The cost of manufacturing the eleventh toy is then: \[ C(11) - C(10) = 156.42 - 152 = 4.42 \] Therefore, the cost of manufacturing the eleventh toy is \( 4.42 \) dollars.

Key Concepts

derivatives

In the world of calculus, derivatives measure how functions change as their inputs change. For example, if we're looking at the cost of manufacturing toys, the derivative of that cost gives us the marginal cost: this tells us how much the cost will change when we produce one more toy.
To find the marginal cost function, we differentiate the given cost function with respect to the number of toys, which in calculus terms is represented as: \( C(x) = 110 + 4x + 0.02x^2 \).
By applying differentiation rules term by term, we find that the marginal cost function is: \( MC(x) = 4 + 0.04x \). This formula lets us calculate the marginal cost for any number of toys produced.

cost functions

A cost function is a mathematical formula that businesses use to determine the cost of producing a certain number of products. In our example, the cost function is: \( C(x) = 110 + 4x + 0.02x^2 \).
This equation consists of three terms:

• 110: the fixed cost, which does not change with the number of toys.
• 4x: the linear cost, which is directly proportional to the number of toys produced.
• 0.02x²: the quadratic cost, which reflects how costs increase at an increasing rate as more toys are produced.

By understanding this, businesses can predict how costs will change and adjust their production strategies. For example, if the total cost for 11 toys is \( C(11) = 156.42 \) dollars, and for 10 toys it is \( C(10) = 152 \) dollars, then the cost of manufacturing the eleventh toy is just \( 156.42 - 152 = 4.42 \) dollars.

manufacturing cost analysis

Manufacturing cost analysis involves breaking down the total cost to understand how each component of the production process contributes to the overall expense. This not only includes physical materials but also fixed costs and variable costs.
In our example, we observed that the cost function involves fixed costs (like rent or salaries), variable costs (that change linearly with production), and other variable costs that increase non-linearly.
Understanding the cost functions allows a company to:

• Control budget more effectively.
• Find the optimal production level to maximize profit.
• Make informed decisions regarding pricing and production strategies.

For instance, knowing the marginal cost at different levels of production enables a company to determine the most cost-effective point of production. When producing 10 toys, the marginal cost is \( MC(10) = 4.4 \) dollars, giving a clear and immediate insight into the additional expense of ramping up production slightly.