Find T, N, and B for the Given Space Curve $$\mathbf { r } ( \mathrm { t } ) = \left( \mathrm { t } ^ { 2 } - 9 \right) \mathbf { i } + ( 2 \mathrm { t } - 9 ) \mathbf { j } + 4 \mathbf { k }$$

question 72

Multiple Choice

Find T, N, and B for the given space curve.
- $$\mathbf { r } ( \mathrm { t } ) = \left( \mathrm { t } ^ { 2 } - 9 \right) \mathbf { i } + ( 2 \mathrm { t } - 9 ) \mathbf { j } + 4 \mathbf { k }$$

• A) \(T = \frac { t } { t ^ { 2 } + 1 } \mathbf { i } + \frac { 1 } { t ^ { 2 } + 1 } j ; N = \frac { 1 } { t ^ { 2 } + 1 } \mathbf { i } - \frac { t } { t ^ { 2 } + 1 } j ; B = - k\)
• B) \(T = \frac { t } { \sqrt { t ^ { 2 } + 1 } } \mathrm { i } + \frac { 1 } { \sqrt { t ^ { 2 } + 1 } } \mathrm { j } ; \mathbf { N } = \frac { 1 } { \sqrt { t ^ { 2 } + 1 } } \mathrm { i } - \frac { \mathrm { t } } { \sqrt { \mathrm { t } ^ { 2 } + 1 } } \mathrm { j } ; \mathbf { B } = - \mathbf { k }\)
• C) \(T = \frac { t } { t ^ { 2 } + 1 } \mathbf { i } + \frac { 1 } { t ^ { 2 } + 1 } j ; N = \frac { 1 } { t ^ { 2 } + 1 } \mathbf { i } - \frac { t } { t ^ { 2 } + 1 } j ; B = k\)
• D) \(T = \frac { t } { t ^ { 2 } + 1 } \mathrm { i } - \frac { 1 } { t ^ { 2 } + 1 } j ; N = - \frac { 1 } { t ^ { 2 } + 1 } i - \frac { t } { t ^ { 2 } + 1 } j ; B = - k\)

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Definitions:

Current Ratio

A liquidity ratio that measures a company's ability to pay short-term obligations with its current assets.

Total Current Assets

The sum of all assets that a company expects to convert into cash, sell, or consume within one year or within the normal operating cycle of the business.

Current Liabilities

Obligations a company is expected to pay within a year, including accounts payable, short-term loans, and other short-term debts.

Common Size Income Statement

An income statement in which each line item is expressed as a percentage of sales, facilitating comparison across companies or periods regardless of size.