For a certain manufacturer, revenue has been increasing but...

For a certain manufacturer, revenue has been increasing but so has the cost of materials and the cost of employee benefits. Suppose revenue can be modeled by $R(t)=10 \sqrt{t},$ the cost of materials by $M(t)=2 t+1,$ and the cost of benefits by $C(t)=0.1 t^{2}+2,$ where $t$ represents the number of months since operations began and outputs are in thousands of dollars. Use this information to complete the following. a. Find the function that represents the total manufacturing costs. b. Find the function that represents how much more the operating costs are than the cost of materials. c. What was the cost of operations in the 10 th month after operations began? d. How much less were the operating costs than the cost of materials in the 10 th month? e. Find the function that represents the profit earned by this company. f. Find the amount of profit earned in the 5 th month and 10 th month. Discuss each result.

For a certain manufacturer, revenue has been increasing but so has the cost of materials and the cost of employee benefits. Suppose revenue can be modeled by $R(t)=10 \sqrt{t},$ the cost of materials by $M(t)=2 t+1,$ and the cost of benefits by $C(t)=0.1 t^{2}+2,$ where $t$ represents the number of months since operations began and outputs are in thousands of dollars. Use this information to complete the following.
a. Find the function that represents the total manufacturing costs.
b. Find the function that represents how much more the operating costs are than the cost of materials.
c. What was the cost of operations in the 10 th month after operations began?
d. How much less were the operating costs than the cost of materials in the 10 th month?
e. Find the function that represents the profit earned by this company.
f. Find the amount of profit earned in the 5 th month and 10 th month. Discuss each result.

Key Concepts

Function Modeling in Economics

This concept involves representing real-world quantities like revenue, material cost, and employee benefits as functions of time. By creating these mathematical models, one can analyze trends and relationships within a business, compare different metrics, and make predictions about future performance.

Arithmetic Operations on Functions

This concept covers the use of addition and subtraction (and other operations) on functions to combine or compare different economic quantities. For instance, total costs can be found by adding separate cost functions, and profit can be modeled by subtracting cost functions from the revenue function.

Function Evaluation

In economic modeling, it is often necessary to determine the value of a function at a particular point in time. This involves substituting a specific value into the function to obtain a numerical output, such as calculating the cost or profit during a certain month.

Profit Function Formulation

This key concept involves deriving a function that represents profit by subtracting the total costs from revenue. Profit functions are critical for assessing the financial health of an operation, as they provide insight into how much revenue exceeds the costs incurred over time.

Transcript

00:01 Okay, so part a wants us to find a total manufacturing cost.

00:04 So i'm going to call that t.

00:08 So t of lowercase t is going to equal our m, which is cost of materials plus the cost of benefits.

00:24 Okay, so the total cost is just going to be 2t plus 1 plus 0 .1t squared plus 2, which is equal to in standard form 0 .1 t squared plus 2t plus 3.

00:52 Part b, by a function that represents how much more the operating costs are than the cost of materials.

01:03 Okay, so let's call it difference.

01:24 So the difference in the cost, it wants to know how much more the operating.

01:31 Creating costs are so that's going to be c f t the cost of benefits minus the cost of the materials okay so that's going to be 0 .1 t squared plus two minus uh two t plus one so it's to be plus two minus one which is going to be plus one uh part c says what was the cost of operations in the 10th month after operations began.

02:17 So that's going to be the total cost.

02:28 Because the party just wants the cost of operations in the 10th month.

02:32 So that's just going to be c of 10 because t was in months.

02:38 So that's just going to be 0 .1 times 10 squared plus two.

02:49 So the total cost and it's in thousands of dollars.

02:56 So 100 times point one is 10.

02:57 So it's going to be $12 ,000.

03:09 Okay, part d.

03:10 One, how much less were the operating costs and the cost of materials in the 10th month? okay, so that's just going to be our difference.

03:21 And so our difference should give us a negative number because it says that materials is greater.

03:26 And so we're just going to flip that negative...