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Bot GPT-4o To determine the depth of the oil in the rectangular tin, we first need to calculate the volume of the engine oil in the cylindrical tin and then use that volume to find the depth in the rectangular tin.
### Step 1: Calculate the volume of oil in the cylindrical tin
The formula for the volume of a cylinder is given by:
\[ V = \pi r^2 h \]
where:
- \( V \) is the volume,
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder.
Given:
- Diameter of the cylindrical tin \( d = 12 \) cm, so radius \( r = \frac{d}{2} = \frac{12}{2} = 6 \) cm,
- Height \( h = 14 \) cm.
Substitute the values into the formula:
\[ V = \pi (6 \text{ cm})^2 (14 \text{ cm}) \]
\[ V = \pi (36 \text{ cm}^2) (14 \text{ cm}) \]
\[ V = \pi (504 \text{ cm}^3) \]
\[ V \approx 3.1416 \times 504 \text{ cm}^3 \]
\[ V \approx 1584 \text{ cm}^3 \]
### Step 2: Use the volume to find the depth in the rectangular tin
The volume of the oil will remain the same when poured into the rectangular tin. The volume formula for a rectangular prism (box) is:
\[ V = l \times w \times h \]
where:
- \( l \) is the length,
- \( w \) is the width,
- \( h \) is the height (or in this case, the depth of the oil, which we need to find).
Given:
- Length \( l = 16 \) cm,
- Width \( w = 11 \) cm.
Substitute the known values and solve for \( h \):
\[ 1584 \text{ cm}^3 = 16 \text{ cm} \times 11 \text{ cm} \times h \]
\[ 1584 \text{ cm}^3 = 176 \text{ cm}^2 \times h \]
\[ h = \frac{1584 \text{ cm}^3}{176 \text{ cm}^2} \]
\[ h = 9 \text{ cm} \]
Thus, the depth of the oil in the rectangular tin is \( 9 \) cm.