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MissPhiladelphia
An apothem is a line drawn from the centerpoint of the polygon to one side of the polygon. There is a formula for area in terms of apothem:

A = (1/2)*(Perimeter)*(Apothem)

The perimeter of the regular hexagon is just the length of one side multiplied with the number of sides. Since a hexagon has 6 sides,

P = 6(15) = 90in

A = 1/2 * 90 * 13
A = 585 square inches

Answer:

Option 1st is correct

585 [tex]\text{in}^2[/tex]

Step-by-step explanation:

Area(A) of a regular hexagon is given by:

[tex]A = \frac{1}{2}P \cdot a[/tex]           ....[1]

where,

P is the perimeter and a is the apothem of the regular hexagon.

As per the statement:

An apothem 13 inches long and a side 15 inches long.

⇒a = 13 inches and side = 15 inches

Perimeter of hexagon(P) = 6s ; where s is the side

⇒P = 6(15) = 90 inches

Substitute the given values in [1] we have;

[tex]A = \frac{1}{2} \cdot 90 \cdot 13[/tex]

Simplify:

A = 585 square inches.

Therefore,  the area of a regular hexagon with an apothem 13 inches long and a side 15 inches long is, [tex]585 in^2[/tex]