Final answer:
To find the height of a cylinder with a known radius and surface area, subtract the area of the bases from the total surface area to get the lateral surface area, and then solve for the height. Using the surface area formula SA = 2πr² + 2πrh, the height is computed to be approximately 9.4 cm.
Step-by-step explanation:
To calculate the height of a cylinder when you know its surface area and radius, you use the surface area formula for a cylinder, SA = 2πr² + 2πrh, where SA is the surface area, r is the radius, and h is the height of the cylinder. Given that the surface area (SA) is 1,220 cm² and the radius (r) is 10 cm, the formula can be rearranged to solve for h (height). The first step is to calculate the area of the circular bases, which is 2πr², then subtract this value from the total surface area to get the lateral surface area (2πrh), and finally divide by (2πr) to solve for h.
First, calculate the area of the circular bases:
Area of one base = πr² = 3.14159 × 10² cm² = 314.159 cm²
Total area of both bases = 2 × 314.159 cm² = 628.318 cm²
Subtract this from the total surface area to find the lateral surface area:
Lateral surface area = SA - area of bases = 1,220 cm² - 628.318 cm² = 591.682 cm²
Finally, solve for the height:
2πrh = 591.682 cm²
h = 591.682 cm² / (2 × 3.14159 × 10 cm)
h ≈ 9.4 cm
Thus, the height of the cylinder is approximately 9.4 cm.