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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!

 Feb 22, 2020
 #1
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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?

 Feb 22, 2020
 #2
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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.

 Feb 22, 2020
 #3
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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).

 Feb 25, 2020
 #4
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 Apr 1, 2020

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