Final answer:
The diameter of the cylinder is approximately 8 cm and the total surface area is approximately 193.624 cm².
Step-by-step explanation:
To find the diameter and total surface area of a cylinder, we first need to calculate its volume. The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height. In this case, the volume of the cylinder is given as 1056 cm³ and the height is 21 cm. Plugging in these values, we can solve for the radius:
1056 = 3.142 × r² × 21
Divide both sides by 3.142 × 21:
r² = 2
Take the square root of both sides:
r = √2
The diameter of the cylinder is twice the radius, so the diameter is 2√2. To find the total surface area, we use the formula A = 2πrh + πr², where A is the total surface area. Plugging in the values, we get:
A = 2 × 3.142 × √2 × 21 + 3.142 × (√2)²
A = 2 × 3.142 × √2 × 21 + 3.142 × 2
A = 2 × 3.142 × √2 × 21 + 6.284
Finally, we can calculate the total surface area:
A = 132.4√2 + 6.284
Simplifying further, we get:
A ≈ 132.4 × 1.414 + 6.284
A ≈ 187.34 + 6.284
A ≈ 193.624
Therefore, the diameter of the cylinder is approximately 8 cm and the total surface area is approximately 193.624 cm².