1. **State the problem:** We have three similar solid cylinders L, M, and P made from the same material. Given masses and surface areas, we need to find the ratio of their heights: height of L : height of M : height of P. 2. **Recall formulas and rules:** - Since the cylinders are similar, their linear dimensions scale by a factor $k$. - Mass is proportional to volume, and volume scales as the cube of the linear scale factor: $m \propto k^3$. - Surface area scales as the square of the linear scale factor: $A \propto k^2$. - Therefore, if $k_L$, $k_M$, and $k_P$ are the scale factors for L, M, and P respectively, then: $$\frac{m_L}{m_M} = \left(\frac{k_L}{k_M}\right)^3$$ $$\frac{A_M}{A_P} = \left(\frac{k_M}{k_P}\right)^2$$ 3. **Find scale factors from mass:** Given $m_L = 64$, $m_M = 125$, $$\frac{64}{125} = \left(\frac{k_L}{k_M}\right)^3$$ Taking cube root: $$\frac{k_L}{k_M} = \sqrt[3]{\frac{64}{125}} = \frac{4}{5}$$ 4. **Find scale factors from surface area:** Given $A_M = 144$, $A_P = 16$, $$\frac{144}{16} = \left(\frac{k_M}{k_P}\right)^2$$ Simplify: $$9 = \left(\frac{k_M}{k_P}\right)^2$$ Taking square root: $$\frac{k_M}{k_P} = 3$$ 5. **Express all scale factors relative to $k_M$:** $$k_L = \frac{4}{5} k_M$$ $$k_P = \frac{k_M}{3}$$ 6. **Since height scales linearly with $k$, the height ratio is:** $$\text{height L} : \text{height M} : \text{height P} = k_L : k_M : k_P = \frac{4}{5} k_M : k_M : \frac{1}{3} k_M$$ Cancel $k_M$: $$\frac{4}{5} : 1 : \frac{1}{3}$$ 7. **Convert to whole numbers by multiplying all terms by 15 (LCM of 5 and 3):** $$\frac{4}{5} \times 15 = 12$$ $$1 \times 15 = 15$$ $$\frac{1}{3} \times 15 = 5$$ 8. **Final height ratio:** $$\boxed{12 : 15 : 5}$$ This means the heights of cylinders L, M, and P are in the ratio 12:15:5.