The expression for the surface area of a cylinder is shown below. Use it to answer Part A, Part B, and Part C. $$ S A=2 \pi\left(\frac{1}{2}\right)^{2}+2 \pi\left(\frac{1}{2}\right)(7) $$ Part A: Find the radius of the cylinder. Part B: Find the height of the cylinder. Part C: Explain what the two portions of the formula $$ 2 \pi\left(\frac{1}{2}\right)^{2} $$ and $$ 2 \pi\left(\frac{1}{2}\right)(7) $$ represent.

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Certainly! Let's break down each part of the problem step by step.

---

### **Given:**

The surface area $$ SA $$ of a cylinder is expressed as:
$$
SA = 2\pi\left(\frac{1}{2}\right)^2 + 2\pi\left(\frac{1}{2}\right)(7)
$$

This formula corresponds to the standard surface area formula for a cylinder:
$$
SA = 2\pi r^2 + 2\pi r h
$$
where:
- $$ r $$ is the radius of the cylinder.
- $$ h $$ is the height of the cylinder.

---

### **Part A: Find the Radius of the Cylinder**

**Identify the Radius ($$ r $$) from the Formula**

Compare the given surface area formula to the standard formula:
$$
2\pi\left(\frac{1}{2}\right)^2 \quad \text{corresponds to} \quad 2\pi r^2
$$

From this comparison:
$$
r = \frac{1}{2}
$$

**Answer:**
The radius of the cylinder is $$ \boxed{\dfrac{1}{2}} $$ units.

---

### **Part B: Find the Height of the Cylinder**

**Identify the Height ($$ h $$) from the Formula**

Again, compare the given formula to the standard one:
$$
2\pi\left(\frac{1}{2}\right)(7) \quad \text{corresponds to} \quad 2\pi r h
$$

We already know $$ r = \frac{1}{2} $$. Plugging this into the equation:
$$
2\pi\left(\frac{1}{2}\right) h = 2\pi\left(\frac{1}{2}\right)(7)
$$
This implies:
$$
h = 7
$$

**Answer:**
The height of the cylinder is $$ \boxed{7} $$ units.

---

### **Part C: Explain the Two Portions of the Formula**

**Breakdown of the Surface Area Formula:**
$$
SA = 2\pi r^2 + 2\pi r h
$$
- **First Term:** $$ 2\pi r^2 $$
  - **Interpretation:** This represents the combined area of the two circular bases (top and bottom) of the cylinder.
  - **Given Expression:** $$ 2\pi\left(\frac{1}{2}\right)^2 $$

- **Second Term:** $$ 2\pi r h $$
  - **Interpretation:** This represents the lateral (curved) surface area of the cylinder—the area around the side.
  - **Given Expression:** $$ 2\pi\left(\frac{1}{2}\right)(7) $$

**Answer:**
- The first portion, $$ 2\pi\left(\dfrac{1}{2}\right)^2 $$, calculates the total area of the cylinder’s two circular bases.
- The second portion, $$ 2\pi\left(\dfrac{1}{2}\right)(7) $$, calculates the area of the curved side (lateral surface) of the cylinder.

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### **Summary of Answers:**

- **Part A:** Radius $$ r = \boxed{\dfrac{1}{2}} $$ units.
- **Part B:** Height $$ h = \boxed{7} $$ units.
- **Part C:** 
  - $$ 2\pi\left(\dfrac{1}{2}\right)^2 $$ represents the area of the two circular bases.
  - $$ 2\pi\left(\dfrac{1}{2}\right)(7) $$ represents the lateral (curved) surface area of the cylinder.
- **Part A:** The radius of the cylinder is $$ \frac{1}{2} $$ units. - **Part B:** The height of the cylinder is 7 units. - **Part C:** - $$ 2\pi\left(\frac{1}{2}\right)^2 $$ is the area of the two circular bases. - $$ 2\pi\left(\frac{1}{2}\right)(7) $$ is the curved surface area of the cylinder.

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