VWXYZ is a rectangle-based pyramid where WX=66 cm and XY=32cm. The vertex is vertically above the center of the base. Given that the slant heights VA and VB are 56 cm and 63 cm respectively, find the total surface area of the pyramid. Additionally, find the height and volume of the pyramid.

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1. Surface area:

The points A and B are the mid-points of WX and XY respectively.

For triangle VWX, the surface area is half the base(66) multiplied by the slant height VA of 56.

For triangle VXY, the surface area is half the base(32) times the slant height of 63.

Add and double the two areas to get the total slant areas, add to the area of the base, a rectangle 66x32, to get the total surface area.

Height of Pyramid:
Drop a perpendicular from V to the centre of the base at point D. Then VAD and VBD are right triangles.
VA=56, AD=XY/2=16, VD=√2880 by Pythagoras theorem.
Similarly,
VB=63, BD=WX/2=33, and again, VD=√(63²-33²)=√2880.

Therefore height of pyramid=√2880.

Volume:
Volume of a pyramid is the area of the base multiplied by (1/3) of the height.

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