The volume V of a solid right circular cylinder is given by 
V = 𝜋r^2h 
 where r is the radius of the cylinder and h is its height. A soda can has inner radius 
r = 1.5 inches,
 height 
h = 9 inches,
 wall thickness 0.02 inches, and top and bottom thickness 0.06 inches. Use linearization to compute the volume, in in^3, of metal in the walls and top and bottom of the can. Give your answer to 2 decimal places.

We are asked to calculate the volume of metal in the walls and top/bottom of a soda can using **linearization**. Let's go step by step.

### Step 1: Volume of the outer can
First, consider the outer volume of the can. The outer radius \( r_{\text{outer}} \) is given by the inner radius plus the thickness of the walls:
\[
r_{\text{outer}} = r + t_{\text{wall}} = 1.5 + 0.02 = 1.52 \, \text{inches}
\]
The height of the outer can is given by the height of the can plus the thickness of the top and bottom:
\[
h_{\text{outer}} = h + 2 \times t_{\text{top}} = 9 + 2 \times 0.06 = 9.12 \, \text{inches}
\]

The volume of the outer cylinder is:
\[
V_{\text{outer}} = \pi (r_{\text{outer}})^2 h_{\text{outer}} = \pi (1.52)^2 (9.12)
\]

### Step 2: Volume of the inner can
The inner can has radius \( r = 1.5 \) inches and height \( h = 9 \) inches. The volume of the inner can is:
\[
V_{\text{inner}} = \pi r^2 h = \pi (1.5)^2 (9)
\]

### Step 3: Volume of the metal in the walls
The volume of the metal in the walls is the difference between the outer and inner volumes:
\[
V_{\text{walls}} = V_{\text{outer}} - V_{\text{inner}}
\]

### Step 4: Volume of the top and bottom
The top and bottom have thickness \( t_{\text{top}} = 0.06 \) inches. The surface area of the top and bottom is:
\[
A_{\text{top}} = A_{\text{bottom}} = \pi r^2 = \pi (1.5)^2
\]
Thus, the volume of the metal in the top and bottom is:
\[
V_{\text{top+bottom}} = 2 \times A_{\text{top}} \times t_{\text{top}} = 2 \times \pi (1.5)^2 \times 0.06
\]

### Step 5: Total volume of metal
The total volume of the metal in the can is the sum of the volume in the walls and the volume in the top and bottom:
\[
V_{\text{metal}} = V_{\text{walls}} + V_{\text{top+bottom}}
\]

Now let's calculate these values.The total volume of metal in the walls and the top and bottom of the soda can is approximately \( 3.43 \, \text{in}^3 \).

Do you have any further questions or need more details?

Here are some related questions:
1. How does the thickness of the walls affect the volume of the metal in the can?
2. What would the volume of metal be if the radius or height of the can were larger?
3. Can this method of linearization be applied to other shapes, such as spheres or cones?
4. How would the result change if the material thickness was different on the top/bottom compared to the sides?
5. What are some real-world implications of reducing the metal thickness for cost or weight savings?

**Tip:** In problems involving small changes, linearization can provide a quick approximation to more complicated functions.

The volume V of a solid right circular cylinder is given by V = 𝜋r^2h where r is the radius of the cylinder and h is its height. A soda can has inner radius r = 1.5 inches, height h = 9 inches, wall thickness 0.02 inches, and top and bottom thickness 0.06 inches. Use linearization to compute the volume, in in^3, of metal in the walls and top and bottom of the can. Give your answer to 2 decimal places.

Learn how to use linearization to compute the volume of metal in the walls and top/bottom of a soda can. This problem applies the cylinder volume formula and geometric principles, with a detailed solution. Suitable for Grades 10-12.

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