Find the Height of a cylinder whose radius is 7 cm and the total surface area is 968 cm$^2$

Question

Accepted Answer

To find the height of the cylinder, we need to use the formula for the total surface area of a cylinder, which is given by:

$$ \text{Total Surface Area} = 2\pi r(h + r) $$

where $ r $ is the radius and $ h $ is the height of the cylinder. We are given that the radius $ r = 7 $ cm and the total surface area is 968 cm$^2$.

Substituting the known values into the formula, we have:

$$ 968 = 2\pi \times 7 \times (h + 7) $$

Simplifying, we get:

$$ 968 = 14\pi (h + 7) $$

To isolate $ h $, divide both sides by $ 14\pi $:

$$ h + 7 = \frac{968}{14\pi} $$

Calculating the right-hand side:

$$ h + 7 = \frac{968}{14 \times 3.1416} $$

$$ h + 7 \approx \frac{968}{43.9824} $$

$$ h + 7 \approx 22 $$

Subtracting 7 from both sides gives:

$$ h \approx 15 $$

Thus, the height of the cylinder is approximately 15 cm, which corresponds to Option 1.

**Explanation for Other Options:**

- **Option 2 (16 cm):** If the height were 16 cm, substituting back into the formula would not yield the total surface area of 968 cm$^2$. The calculation would result in a larger surface area, thus making this option incorrect.

- **Option 3 (12 cm):** Similarly, if the height were 12 cm, the total surface area calculated would be less than 968 cm$^2$, which does not match the given condition, making this option incorrect.

- **Option 4 (20 cm):** With a height of 20 cm, the total surface area would exceed 968 cm$^2$, thus not satisfying the condition provided in the problem, making this option incorrect.

Therefore, based on the calculations and logical reasoning, Option 1 (15 cm) is the correct answer.