Find the Length of the Curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 3 }$

question 46

Multiple Choice

Find the length of the curve $\mathbf { r } ( t ) = 2 t \mathbf { i } + t ^ { 2 } \mathbf { j } + \ln t \mathbf { k } , 1 \leq t \leq e ^ { 3 }$ .
Select the correct answer.

• A) \(e ^ { 6 }\)
• B) \(e ^ { 3 }\)
• C) \(e ^ { 6 } + 2\)
• D) \(e ^ { 3 } + 2\)

---

Definitions:

Profit-Maximizing Price

The profit-maximizing price is the optimal price a firm can set for its product or service to achieve the highest possible profit, considering the balance between price, demand, and production costs.

Fixed Costs

Expenses that do not change in proportion to the activity of a business, such as rent, salaries, and insurance.

Monopolistically Competitive

A market scenario where various companies offer products that are substitutes but different enough to give the companies some price-setting power.

Profit-Maximizing

The process or strategy of adjusting production and sale practices to achieve the highest possible profit.