Now, suppose the price of a brand new machine is A=$ 3,000. This is the fixed cost. The total depreciation you answered in (a) can be also considered as a cost. Add these two costs to form the total cost that incur during the time interval (0, t). Graph this function. Label the horizontal axis as "month" and the vertical axis as "total cost". Hand (b) drawing is OK! But, make sure the concavity is correct. Label the intercept with a specific dollar amount!
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
Please answer PART B. Show steps clearly and no cursive if possible.
![### WA-4.1: Optimal Time for Copy Machine Overhaul
**Scenario:**
Imagine you are running a copy shop with a single copy machine used 24 hours daily. Over time, the machine wears out, requiring periodic replacement with a new one. This guide focuses on finding the optimal time between overhauls, aiming to achieve the minimum monthly cost.
**Cost Considerations:**
1. **Fixed cost:** The price of replacing the old copy machine.
2. **Monthly depreciation:** The machine’s value depreciation every month, which increases as the machine gets older.
**Tasks:**
**(a)**
**Given:** The machine's depreciation rate is \( f(s) = \frac{100}{\sqrt{s}} \) dollars per month, where \( s \) is the machine’s age in months.
**Task:** Find the total depreciation incurred during the first \( t \) months after an overhaul. The answer should be a function of \( t \).
**(b)**
**Given:** The price of a new machine is \( A = \$3,000 \).
**Task:** Calculate the total depreciation during the interval \([0, t]\) and add it to the fixed cost to get the total cost.
**Graph:** Sketch this total cost function. Use "month" for the horizontal axis and "total cost" for the vertical axis. Label the intercept with a specific dollar amount and ensure the graph shows the correct concavity.
**(c)**
**Task:** From a copy shop owner’s perspective, explain why "Total Cost" is not the main factor to consider. Identify what should be minimized.
**(d)**
**Given:** Let \( C \) denote the **Average Monthly Cost** \( C(t) \).
**Task:** Define and write the average cost function \( C = C(t) \) for \( t \geq 0 \). This function represents the average monthly costs over the interval \([0, t]\). Refer to the textbook for the exact function definition if needed.
**(e)**
**Task:** Determine the optimal "stopping" time \( T \) to minimize the function \( C(t) \).
1. **Critical Number:** Find the exact value of \( T \).
2. **Behavior of \( C \):** Show \( C \) is decreasing on \( 0 < t < T \) and increasing on \( T](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28092663-10ce-474b-a101-95a46deb92bf%2F86d7ef42-5092-4280-a241-4daee8ae7d3e%2Fk4917dv.png&w=3840&q=75)
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