Final answer:
To find the equations of the sides of the pentagon and calculate its area, each line segment's equation is derived and the area is determined by integrating the difference of the upper and lower bounds' functions for each section. Without the specific value of P, we are unable to provide the exact equations or verify the given area solutions.
Step-by-step explanation:
To find the equations of all the five sides of the given pentagon and to determine the area, we need to use the coordinates provided to derive each line segment's linear equation and then integrate with respect to the x-axis to find the total area enclosed by the pentagon.
Step 1: Find the Equations of the Sides
Let's start by finding the equation of each side:
Side 3: Joining (P +6,5) to (P + 3, 10) - Use the two-point formula or slope-intercept form to find the equation.
Side 5: Joining (P,5) to (P,0) - This is another vertical line segment, where the x-value remains P for all y.
Note: Without specific values for P, actual equations cannot be provided. We would only be able to give the equations in terms of P.
Step 2: Calculate the Area
To find the area of the pentagon using integration, we would subdivide it into recognizable shapes (such as triangles and rectangles) or integrate the function describing the upper bound of the pentagon minus the function describing the lower bound of the Pentagon for each relevant section.
Since the correct equations and the exact value of P are not provided, we cannot calculate the precise area of the pentagon or confirm the given equations A, B, C, or D.