1. **Stating the problem:** A manufacturer must increase the height of a rectangular wooden door by $\frac{1}{6}$ of its original height while keeping the thickness the same. To maintain the same material cost, the width must be reduced. We need to find the ratio between the new width and the original width. 2. **Define variables:** Let the original height be $h$ and the original width be $w$. 3. **New dimensions:** - New height = $h + \frac{1}{6}h = \frac{7}{6}h$ - New width = $w_{new}$ (unknown) 4. **Volume (material) preservation:** Since thickness is constant, volume is proportional to height $\times$ width. Original volume $V = h \times w$ New volume $V_{new} = \frac{7}{6}h \times w_{new}$ To keep the cost (volume) the same: $$h \times w = \frac{7}{6}h \times w_{new}$$ 5. **Solve for $w_{new}$:** $$w = \frac{7}{6} w_{new} \implies w_{new} = \frac{6}{7} w$$ 6. **Ratio of new width to original width:** $$\frac{w_{new}}{w} = \frac{6}{7}$$ **Final answer:** The ratio is $\boxed{\frac{6}{7}}$ which corresponds to option c.