Final answer:
To find the surface area of the cylinder gift box, we need to find the area of the two circular bases and the curved surface area. The formula to find the surface area involves the radius and height of the cylinder.
Step-by-step explanation:
To find the surface area of the cylinder, we need to find the area of the two circular bases and the curved surface area. Let's start with the area of the circular bases.
The radius of the cylinder gift box is given as 3X + 1 inches, so the radius of each circular base is also 3X + 1 inches. The formula to find the area of a circle is A = πr², where A is the area and r is the radius. Plugging in the given radius, the area of each circular base is A = π(3X + 1)² square inches.
The height of the gift box is given as twice the radius. So, the height is 2(3X + 1) = 6X + 2 inches.
The curved surface area of the cylinder is equal to the circumference of the base multiplied by the height. The formula to find the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. Plugging in the given radius, the circumference is C = 2π(3X + 1) inches. Multiplying by the height, the curved surface area is (2π(3X + 1))(6X + 2) square inches.
To find the total surface area, we add the areas of the circular bases and the curved surface area. So, the surface area of the cylinder gift box in standard form is:
A = 2(π(3X + 1)²) + (2π(3X + 1))(6X + 2) square inches.