Final answer:
The lateral area and surface area of a cylinder can be computed using its circumference and height. With the given circumference of 97 cm, the estimated lateral area is approximately 2992.3651 cm² and the surface area is approximately 3961.0659 cm².
Step-by-step explanation:
The question deals with finding the lateral area and the surface area of a right cylinder when the circumference and the height-to-radius ratio are given. The circumference (C) of a cylinder is given by the formula C = 2πr, where r is the radius. Given C = 97 cm, we can solve for r:
r = C / (2π) = 97 / (2π) = 15.4318 cm (approximately),
Since the height is twice the radius,
h = 2r = 2 × 15.4318 cm = 30.8636 cm (approximately).
The lateral area (LA) of the cylinder is the perimeter of the base times the height, given by the formula LA = 2πrh. Plugging in the values, we find:
LA = 2π × 15.4318 cm × 30.8636 cm = 2992.3651 cm² (approximately).
The surface area (SA) of a cylinder is calculated by adding the areas of the two bases to the lateral area, SA = 2πr² + LA. Therefore,
SA = 2π × (15.4318 cm)² + 2992.3651 cm² = 3961.0659 cm² (approximately).