1. **Problem Statement:** Calculate the total surface area of a frustum of a right pyramid with rectangular base ABCD where $AB=32$ cm, $BC=24$ cm. The top face EFGH is parallel to ABCD and located three-quarters up the vertical height of the original pyramid. Given $EF=8$ cm, $FG=6$ cm, and the slant height of the original pyramid is 44 cm. 2. **Understanding the frustum:** The frustum is formed by slicing the original pyramid parallel to its base at height $\frac{3}{4}$ of the total height. The top face EFGH is similar to the base ABCD but scaled down. 3. **Surface area components:** The total surface area of the frustum includes: - Area of the bottom base ABCD - Area of the top face EFGH - Lateral surface area (the trapezoidal faces connecting the two bases) 4. **Calculate areas of bases:** - Bottom base area $= AB \times BC = 32 \times 24 = 768$ cm$^2$ - Top base area $= EF \times FG = 8 \times 6 = 48$ cm$^2$ 5. **Find the scale factor:** Since EFGH is three-quarters up, the linear scale factor from base to top is $k = 1 - \frac{3}{4} = \frac{1}{4}$ (because the top is smaller and at 3/4 height, the similarity ratio is $\frac{1}{4}$). 6. **Calculate the vertical height $h$ of the original pyramid:** The slant height $l = 44$ cm is the length along the face from base to apex. The slant height relates to the vertical height $h$ and half the diagonal of the base. - Diagonal of base $d = \sqrt{32^2 + 24^2} = \sqrt{1024 + 576} = \sqrt{1600} = 40$ cm - Half diagonal $= 20$ cm Using Pythagoras in the triangular face: $$ l^2 = h^2 + (\text{half diagonal})^2 $$ $$ 44^2 = h^2 + 20^2 $$ $$ 1936 = h^2 + 400 $$ $$ h^2 = 1536 $$ $$ h = \sqrt{1536} \approx 39.19 \text{ cm} $$ 7. **Calculate the slant height of the frustum $l_f$:** Since the frustum is the top $\frac{1}{4}$ of the height removed, the vertical height of the frustum is $h_f = \frac{1}{4} h = 9.80$ cm. The slant height of the frustum is the difference between the original slant height and the slant height of the smaller top pyramid: - Smaller top pyramid slant height $l_t = k \times l = \frac{1}{4} \times 44 = 11$ cm - Frustum slant height $l_f = l - l_t = 44 - 11 = 33$ cm 8. **Calculate the lateral surface area:** The lateral surface area of the frustum is the sum of the areas of the trapezoidal faces. Each face is a trapezium with parallel sides equal to the corresponding edges of the bases and height equal to the slant height of the frustum. - For sides parallel to AB and EF: $$ \text{Average length} = \frac{AB + EF}{2} = \frac{32 + 8}{2} = 20 $$ - For sides parallel to BC and FG: $$ \text{Average length} = \frac{24 + 6}{2} = 15 $$ Lateral surface area: $$ L = (\text{average length}_1 + \text{average length}_2) \times l_f = (20 + 15) \times 33 = 35 \times 33 = 1155 \text{ cm}^2 $$ 9. **Calculate total surface area:** $$ \text{Total surface area} = \text{bottom area} + \text{top area} + \text{lateral area} = 768 + 48 + 1155 = 1971 \text{ cm}^2 $$ 10. **Final answer:** The total surface area of the frustum is approximately **1971 cm$^2$** to the nearest whole number.