1. **Stating the problem:** We are given cubes with edge lengths $e$ and need to complete the table for total surface area (T.A.) and volume (V). 2. **Formulas for cubes:** - Total surface area: $$\text{T.A.} = 6e^2$$ - Volume: $$V = e^3$$ 3. **Given values and unknowns:** - For $e=3$, find T.A. and V. - For $e = e$ (symbolic), find T.A. and V. - For $e=2x$, find T.A. and V. - Given T.A. = 150, find $e$. - Given $V=1000$ and $V=64$, find $e$. 4. **Calculations:** **For $e=3$:** - $$\text{T.A.} = 6 \times 3^2 = 6 \times 9 = 54$$ - $$V = 3^3 = 27$$ **For $e = e$ (symbolic):** - $$\text{T.A.} = 6e^2$$ - $$V = e^3$$ **For $e=2x$:** - $$\text{T.A.} = 6(2x)^2 = 6 \times 4x^2 = 24x^2$$ - $$V = (2x)^3 = 8x^3$$ **Given T.A. = 150:** - $$6e^2 = 150$$ - Divide both sides by 6: $$\cancel{6}e^2 = \cancel{6}25$$ - $$e^2 = 25$$ - Taking square root: $$e = 5$$ **Given $V=1000$:** - $$e^3 = 1000$$ - Taking cube root: $$e = 10$$ **Given $V=64$:** - $$e^3 = 64$$ - Taking cube root: $$e = 4$$ 5. **Completed table:** | $e$ | 3 | $e$ | $2x$ | | | | |---|---|---|---|---|---|---| | T.A. | 54 | $6e^2$ | $24x^2$ | 150 (given) | | | | V | 27 | $e^3$ | $8x^3$ | | 1000 | 64 | 6. **Summary:** - For $e=3$, T.A. = 54, V = 27. - For $e=e$, T.A. = $6e^2$, V = $e^3$. - For $e=2x$, T.A. = $24x^2$, V = $8x^3$. - If T.A. = 150, then $e=5$. - If $V=1000$, then $e=10$. - If $V=64$, then $e=4$. **Final answer:** $$\boxed{\begin{cases} e=3: & \text{T.A.}=54, V=27 \\ e=e: & \text{T.A.}=6e^2, V=e^3 \\ e=2x: & \text{T.A.}=24x^2, V=8x^3 \\ \text{T.A.}=150: & e=5 \\ V=1000: & e=10 \\ V=64: & e=4 \end{cases}}$$