1. **Problem statement:** We have a square-based pyramid with base side length 24 cm and triangular faces that are isosceles triangles with base 24 cm and equal sides forming a 55° angle at the base. 2. **Find the height $h$ of the pyramid:** - The base is a square with side length $24$ cm. - The apex height $h$ is the perpendicular height from the apex to the center of the base. - The slant height $a$ is the height of the triangular face. - The angle between the base edge and the slant height is $55^\circ$. 3. **Calculate the slant height $a$ of the triangular face:** - The triangle PQR is isosceles with base $24$ cm and equal sides $a$. - The apex angle is $55^\circ$, so half the base angle is $55^\circ$. - Using right triangle formed by half the base and slant height: $$\cos 55^\circ = \frac{12}{a} \implies a = \frac{12}{\cos 55^\circ}$$ $$a = \frac{12}{0.5736} \approx 20.92 \text{ cm}$$ 4. **Calculate the height $h$ of the pyramid:** - The height $h$ is the vertical height from the apex to the base center. - The base diagonal is $24\sqrt{2}$ cm, so half the diagonal is $12\sqrt{2}$ cm. - Using the right triangle formed by $h$, $a$, and half the diagonal: $$h = \sqrt{a^2 - (12\sqrt{2})^2}$$ $$h = \sqrt{20.92^2 - (12\times1.414)^2} = \sqrt{437.7 - 288} = \sqrt{149.7} \approx 12.23 \text{ cm}$$ 5. **Calculate the volume $V$ of the pyramid:** - Volume formula for a pyramid: $$V = \frac{1}{3} \times \text{base area} \times h$$ - Base area: $$24 \times 24 = 576 \text{ cm}^2$$ - Volume: $$V = \frac{1}{3} \times 576 \times 12.23 = 2304 \text{ cm}^3 \approx 2350 \text{ cm}^3 \text{ (nearest integer)}$$ 6. **Calculate the percentage increase in volume when dimensions increase by 10%:** - New side length: $$24 \times 1.1 = 26.4 \text{ cm}$$ - New height: $$12.23 \times 1.1 = 13.453 \text{ cm}$$ - New volume: $$V_{new} = \frac{1}{3} \times 26.4^2 \times 13.453 = \frac{1}{3} \times 696.96 \times 13.453 = 3125.5 \text{ cm}^3$$ - Percentage increase: $$\frac{3125.5 - 2304}{2304} \times 100 = \frac{821.5}{2304} \times 100 \approx 35.65\%$$ **Final answers:** - Height $h \approx 12.23$ cm - Volume $\approx 2350$ cm$^3$ - Percentage increase in volume $\approx 35.7\%$