A 3-Gram Drop of Thick Red Paint Is Added to a Large

question 67

Multiple Choice

A 3-gram drop of thick red paint is added to a large can of white paint.A red disk forms and spreads outward, growing lighter at the edges.Since the amount of red paint stays constant through time, the density of the red paint in the disk must vary with time.Suppose that its density p in gm/cm² is of the form $p=k(t) f(r)$ for some functions k(t) of time and f(r) of the distance to the center of the disk.Let R(t) be the radius of the disk at time t.Which of the following equations expresses the fact that there are 3 grams of red paint in the disk?

• A) \(3=k(t)\)\(\int_{0}^{R(t)}\)\(f(r) 2 \pi r d r\)
• B) \(R(t)=k(t) \int_{0}^{3} f(r) 2 \pi r d r\)
• C) \(3=f(r)\)\(\int_{0}^{p(r)}\)\(k(t) 2 \pi r d r\)
• D) \(p(r, t)=k(t)\)\(\int_{0}^{3} f(r) 2 \pi r d r\)

---

Definitions:

Differentiated Product

A product that is distinguished from similar products offered by competitors through variations in design, functionality, or quality, making it more appealing to a specific segment of consumers.

Monopolistically Competitive Firm

A company that operates in a market structure characterized by many firms selling products that are similar but not identical, allowing for some degree of market power and product differentiation.

Market Share

The market share that a certain company or product possesses within the industry.

Product

Any item or service offered for sale, capable of satisfying a customer's need or want.