1. **State the problem:** We have two hemispheres and a right prism. The total volume of the two hemispheres equals the volume of the prism. We need to find the value of $p$. 2. **Given data:** - Diameter of each hemisphere = 21 cm, so radius $r = \frac{21}{2} = 10.5$ cm. - Prism base length = 26 cm. - Prism base height = 12 cm. - Prism cross-section height = 16 cm. - Prism slant height = $p$ cm (to be found). - Use $\pi = \frac{22}{7}$. 3. **Volume formulas:** - Volume of a hemisphere: $V = \frac{2}{3} \pi r^3$. - Volume of prism: $V = \text{base area} \times \text{height}$. 4. **Calculate volume of two hemispheres:** $$ V_{2 hemispheres} = 2 \times \frac{2}{3} \pi r^3 = \frac{4}{3} \pi r^3 $$ Substitute $r=10.5$: $$ V_{2 hemispheres} = \frac{4}{3} \times \frac{22}{7} \times (10.5)^3 $$ Calculate $10.5^3$: $$ 10.5^3 = 10.5 \times 10.5 \times 10.5 = 1157.625 $$ So, $$ V_{2 hemispheres} = \frac{4}{3} \times \frac{22}{7} \times 1157.625 $$ 5. **Calculate volume of prism:** Base area = base length $\times$ base height = $26 \times 12 = 312$ cm$^2$. Height of prism = $p$ cm. Volume of prism: $$ V_{prism} = 312 \times p $$ 6. **Set volumes equal:** $$ \frac{4}{3} \times \frac{22}{7} \times 1157.625 = 312p $$ Calculate left side: $$ \frac{4}{3} \times \frac{22}{7} = \frac{88}{21} \approx 4.190476 $$ Multiply: $$ 4.190476 \times 1157.625 \approx 4849.5 $$ So, $$ 4849.5 = 312p $$ 7. **Solve for $p$:** $$ p = \frac{4849.5}{312} \approx 15.54 $$ **Final answer:** $$ p \approx 15.54 \text{ cm} $$