Using Green's Theorem, Calculate the Area of the Indicated Region

question 139

Multiple Choice

Using Green's Theorem, calculate the area of the indicated region.
-The astroid $\mathbf { r } ( \mathrm { t } ) = \left( 5 \cos ^ { 3 } \mathrm { t } \right) \mathbf { i } + \left( 5 \sin ^ { 3 } \mathrm { t } \right) \mathbf { j } , 0 \leq \mathrm { t } \leq 2 \pi$

• A) \(\frac { 75 } { 8 } \pi\)
• B) \(\frac { 75 } { 4 } \pi\)
• C) \(2 \pi\)
• D) \(\frac { 75 } { 16 } \pi\)

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Definitions:

Contribution Margin Ratio

The ratio of contribution margin to sales revenue, indicating the percentage of each sales dollar available to cover fixed costs and provide profit.

Fixed Expenses

Costs that do not fluctuate with changes in production level or sales volume, such as rent, salaries, and insurance.

Variable Costing

An accounting method that includes only variable production costs (direct materials, direct labor, and variable manufacturing overhead) in product costs.

Fixed Manufacturing Overhead

is the sum of all the production costs that do not change with the level of output, including costs like factory rent, salaries of permanent employees, and depreciation of factory equipment.