Subjects geometry

Rectangular Prism 55268E

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1. **State the problem:** We are given a rectangular prism with a square front face of side length 6 cm. 2. **Identify the dimensions:** The front face is a square with sides 6 cm each, so the area of the front face is $6 \times 6 = 36$ cm$^2$. 3. **Given information:** The total surface area is 216 cm$^2$. 4. **Find the missing dimension:** Let the depth (length) of the prism be $x$ cm. 5. **Surface area formula for a rectangular prism:** $$\text{Surface Area} = 2(lw + lh + wh)$$ where $l$, $w$, and $h$ are the length, width, and height. 6. **Apply the formula:** Since the front face is a square, let $l = h = 6$ cm and $w = x$ cm. $$216 = 2(6 \times 6 + 6 \times x + 6 \times x)$$ 7. **Simplify inside the parentheses:** $$216 = 2(36 + 6x + 6x) = 2(36 + 12x)$$ 8. **Distribute the 2:** $$216 = 72 + 24x$$ 9. **Isolate $x$:** $$216 - 72 = 24x$$ $$144 = 24x$$ 10. **Divide both sides by 24:** $$\cancel{24}x = \frac{144}{\cancel{24}}$$ $$x = 6$$ 11. **Conclusion:** The depth of the prism is 6 cm. 12. **Final answer:** The prism is a cube with side length 6 cm, and the surface area is 216 cm$^2$ as given. **Answer:** $6.0$ cm (to the nearest tenth)
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