1) Let $APQRS$ be a pyramid, where the base $PQRS$ is a square of side length $20$. The total surface area of pyramid $APQRS$ (including the base) is $600$. Let $W$, $X$, $Y$, and $Z$ be the midpoints of $\overline{AP}$, $\overline{AQ}$, $\overline{AR}$, and $\overline{AS}$, respectively. Find the total surface area of frustum $PQRSWXYZ$ (including the bases).
2) In a certain rectangular prism, the total length of all the edges is $40,$ and the total surface area is $48.$ Find the length of the diagonal connecting one corner to the opposite corner.
3) In a certain regular square pyramid, all of the edges have length $12$. Find the volume of the pyramid.
I'm down with the flu and haven't had time to do these.. Any help is appreciated!
And please help with these problems....I need help badly!
4) The perimeter of the cross-sectional triangle of a prism is $45 \text{ cm}$, the radius of the incircle of the triangle is $8 \text{ cm}$ and the volume of the prism is $900 \text{ cm}^3$. What is the length of the prism?
5) A rectangular prism with length $l$, width $w$, and height $h$ has the property that $l + w + h = 11$ and $l^2 + w^2 + h^2 = 59$ What is the surface area of the prism?
6) A square-based pyramid of height $3$ cm and base length $2$ cm is set on its square base. Water is poured through an infinitesimally small hole in the top of the pyramid to a height of $1$ cm measured from the base, and then the hole is sealed. If the pyramid is turned upside-down, what will be the new height of the water?
+7 Ah these are AOPS questions.
Hint for nr. 3.
We know that the 4 sides are equilateral triangles, and then we can find slant height. Using pythag, we can then find the height of the pyramid.
1) The surface area is 20 + 600/4 = 170.
2) The length of the diagonal is 2*sqrt(17).
3) The height of the pyramid works out to 4*sqrt(3), so the volume is 1/3*4*sqrt(3)*12^2 = 192*sqrt(3).
