A regular pentagon box has a base perimeter (P) of 5b^2− 10b− 15 units and a height of b units. The area of a regular pentagon is A = 1/2 aP, where a = apothem and P = perimeter. If the apothem of the base is 2b − 4 units, what is the surface area of the pentagon box?

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A regular pentagon box has a base perimeter (P) of 5b^2− 10b− 15 units and a height of b units. The area of a regular pentagon is A = 1/2 aP, where a = apothem and P = perimeter. If the apothem  of the base is 2b − 4 units, what is the surface area of the pentagon box?

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Calculate Base Area: Calculate the area of the base of the pentagon. We are given the perimeter of the base (P) as 5b^2− 10b− 15 units and the apothem (a) as 2b − 4 units. The formula for the area of a regular pentagon is A = 1/2 aP. Let's plug in the values to find the area of the base. A = 1/2 * (2b − 4) * (5b^2− 10b− 15)Simplify Base Area: Simplify the expression for the area of the base. A = 1/2 * (2b − 4) * (5b^2− 10b− 15) A = (b − 2) * (5b^2− 10b− 15) Now, distribute (b − 2) across (5b^2− 10b− 15). A = (b − 2) * 5b^2 - (b − 2) * 10b - (b − 2) * 15Distribute Base Area: Perform the distribution to find the area of the base. A = 5b^3 - 10b^2 - 15b - 10b^2 + 20b + 30 Now, combine like terms. A = 5b^3 - 20b^2 + 5b + 30Calculate Lateral Surface Area: Calculate the lateral surface area of the pentagon box. The lateral surface area of a prism is the perimeter of the base times the height. Since we have a pentagon box, we will use the given perimeter and height. Lateral Surface Area = P * height Lateral Surface Area = (5b^2− 10b− 15) * bSimplify Lateral Surface Area: Simplify the expression for the lateral surface area. Lateral Surface Area = 5b^3 - 10b^2 - 15bCalculate Total Surface Area: Calculate the total surface area of the pentagon box. The total surface area is the sum of the area of the base and the lateral surface area. Since the pentagon box has two bases, we need to double the area of the base. Total Surface Area = 2 * Area of base + Lateral Surface Area Total Surface Area = 2 * (5b^3 - 20b^2 + 5b + 30) + (5b^3 - 10b^2 - 15b)Simplify Total Surface Area: Simplify the expression for the total surface area. Total Surface Area = 10b^3 - 40b^2 + 10b + 60 + 5b^3 - 10b^2 - 15b Now, combine like terms. Total Surface Area = 15b^3 - 50b^2 - 5b + 60

A regular pentagon box has a base perimeter $P$ of $5b^2 - 10b - 15$ units and a height of $b$ units. The area of a regular pentagon is $A = \frac{1}{2} aP$ , where $a =$ apothem and $P =$ perimeter. If the apothem of the base is $2b - 4$ units, what is the surface area of the pentagon box?

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