1. **Problem statement:** A photograph originally measures 16 cm by 11 cm. It is cropped by removing the same width $x$ cm from the top and left side, reducing the area by 72 cm$^2$. We need to find the dimensions of the cropped photograph. 2. **Formula and explanation:** The original area is $$16 \times 11 = 176 \text{ cm}^2.$$ After cropping, the new dimensions are $$(16 - x)$$ cm (width) and $$(11 - x)$$ cm (height). The new area is $$(16 - x)(11 - x)$$. The area is reduced by 72 cm$^2$, so: $$176 - (16 - x)(11 - x) = 72$$ 3. **Set up the equation:** $$176 - (16 - x)(11 - x) = 72$$ Simplify: $$(16 - x)(11 - x) = 176 - 72 = 104$$ 4. **Expand the product:** $$ (16 - x)(11 - x) = 16 \times 11 - 16x - 11x + x^2 = 176 - 27x + x^2 $$ Set equal to 104: $$176 - 27x + x^2 = 104$$ 5. **Bring all terms to one side:** $$x^2 - 27x + 176 - 104 = 0$$ $$x^2 - 27x + 72 = 0$$ 6. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{27 \pm \sqrt{(-27)^2 - 4 \times 1 \times 72}}{2} = \frac{27 \pm \sqrt{729 - 288}}{2} = \frac{27 \pm \sqrt{441}}{2} = \frac{27 \pm 21}{2}$$ Two solutions: $$x = \frac{27 + 21}{2} = 24$$ $$x = \frac{27 - 21}{2} = 3$$ 7. **Check for valid solution:** $x=24$ is invalid because it is larger than the original dimensions. So, $x=3$ cm. 8. **Find cropped dimensions:** Width: $$16 - 3 = 13 \text{ cm}$$ Height: $$11 - 3 = 8 \text{ cm}$$ **Final answer:** The cropped photograph measures 13 cm by 8 cm.