Use differentials to estimate the amount of metal (in cm3) in a closed cylindrical can that is 22 cm high and 8 cm in diameter if the metal in the top and the bottom is 0.4 cm thick and the metal in the sides is 0.05 cm thick. (Round your answer to two decimal places.)

We are asked to estimate the volume of metal in a cylindrical can using differentials. Here's the process to solve it:

### Given Information:
- Height of the can, \( h = 22 \, \text{cm} \)
- Diameter of the can, \( d = 8 \, \text{cm} \), so the radius, \( r = \frac{d}{2} = 4 \, \text{cm} \)
- Thickness of the metal in the top and bottom: \( 0.4 \, \text{cm} \)
- Thickness of the metal in the sides: \( 0.05 \, \text{cm} \)

### Step 1: Metal in the Top and Bottom
Each of the top and bottom surfaces is a disk, and we can estimate the volume of metal in one disk as the product of the area of the disk and the thickness of the metal.

- The area of a disk is \( A = \pi r^2 \).
- The volume of the metal in one disk is approximately:
  \[
  V_{\text{disk}} = \pi r^2 \times \text{thickness}
  \]
  Substituting values for the top and bottom:
  \[
  V_{\text{top/bottom}} = \pi (4)^2 \times 0.4 = \pi \times 16 \times 0.4 = 20.106 \, \text{cm}^3
  \]
  Since there are two disks (top and bottom), the total volume of metal for the top and bottom is:
  \[
  V_{\text{top+bottom}} = 2 \times 20.106 = 40.21 \, \text{cm}^3
  \]

### Step 2: Metal in the Sides
The side of the can is a cylindrical surface, and we can estimate the volume of metal in the side as the product of the surface area of the cylinder and the thickness of the metal.

- The surface area of the side is \( A_{\text{side}} = 2\pi r h \).
- The volume of metal in the side is approximately:
  \[
  V_{\text{side}} = 2\pi r h \times \text{thickness}
  \]
  Substituting values:
  \[
  V_{\text{side}} = 2\pi \times 4 \times 22 \times 0.05 = 27.646 \, \text{cm}^3
  \]

### Step 3: Total Volume of Metal
The total volume of metal is the sum of the volumes from the top, bottom, and side:
\[
V_{\text{total}} = V_{\text{top+bottom}} + V_{\text{side}} = 40.21 + 27.646 = 67.86 \, \text{cm}^3
\]

Thus, the total amount of metal in the cylindrical can is approximately **67.86 cm³**.

Would you like more details or have any questions?

### Related Questions:
1. How would the volume change if the height of the can was increased by 5 cm?
2. What if the thickness of the sides was doubled to 0.1 cm? How would that affect the total volume?
3. How can we apply this method to estimate the volume of metal in other shapes like cones or spheres?
4. What is the relationship between the surface area and the volume of metal in this problem?
5. How would the result change if the diameter was changed but the height remained the same?

### Tip: 
When using differentials, approximations are based on small changes in the dimensions. Ensure the thicknesses are small relative to the overall size of the object for accurate results.

Learn how to estimate the volume of metal in a cylindrical can that is 22 cm high and 8 cm in diameter using differentials. The top and bottom have a thickness of 0.4 cm, and the sides are 0.05 cm thick. Step-by-step calculations show how to apply volume and surface area formulas to solve this problem. Suitable for high school students.

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