To find the height of the second cylinder given that it has the same ratio of diameter to height as the first cylinder, we first need to determine the ratio for the first cylinder.
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Calculate the diameter and height of the first cylinder:
- Diameter: \(d_1 = 6\) inches
- Height: \(h_1 = 14\) inches
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Calculate the ratio of diameter to height for the first cylinder: \[ \text{Ratio} = \frac{d_1}{h_1} = \frac{6}{14} = \frac{3}{7} \]
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Let the height of the second cylinder be \(h_2\). The diameter of the second cylinder is given as:
- Diameter: \(d_2 = 3\) inches
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Using the same ratio of diameter to height for the second cylinder: \[ \frac{d_2}{h_2} = \frac{3}{7} \]
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Substituting the diameter of the second cylinder into the ratio equation: \[ \frac{3}{h_2} = \frac{3}{7} \]
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Cross-multiply to solve for \(h_2\): \[ 3 \cdot 7 = 3 \cdot h_2 \] \[ 21 = 3h_2 \]
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Divide both sides by 3: \[ h_2 = \frac{21}{3} = 7 \text{ inches} \]
Thus, the height of the second cylinder is \( \boxed{7} \) inches.
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