area and volume problems

agapela

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Jul 6, 2015
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I am an older student preparing for a Math competency exam. I have three geometry questions I cannot figure out, and am hoping someone can walk me through them/explain how they are solved.

First:
A rectangular prism has a length of 9 units and a width of 4 units. A triangular prism has a base area of 36 square units. If the two prisms have equal volumes, which of the following statements must be true of the two prisms?

Correct Answer: They have the same height.

Second:
A rectangle section of a park has a length l and a width w. The semicircular sections at each of its ends are made of concrete and the shaded portion is planted with grass. Which of the following formulas represents A, the area of the grassy region?
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Correct Answer: lw- (pi x w)2/4

Third:
A tent in the shape of a square pyramid has a floor that measures 6 feet on each side. The slant height of the tent is 8 ft. How much material is needed to make the tent including the floor?
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Correct Answer: 132 square feet

Any help would be GREATLY APPRECIATED!!!!! :p
 

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For the first problem, you'd be best served by drawing a picture of what you know. My poorly-drawn picture is thus:
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You're given the length and width of the rectangular prism. I've labelled the given values, and labelled the unknown heights as h. Similarly, you're given the area of the base of the triangular prism. Take a look at the areas of the areas which are shaded light blue. What do you notice about them? Then, if the volumes of the prisms are the equal, what can you say about their heights?

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For number two, the only thing I notice is that the problem never states that the two semi-circles are the same size, but assumes they are. In any case, the problem asks you how much grass you need to fill the purplish (shaded) area. What does that tell you about the semi-circular areas that are not shaded? From there, can you use the volume formulas of various shapes to figure out the shaded area?

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For number three, you're essentially being asked to find the surface area of the tent. You're told the tent is in the shape of a pyramid. And you can divide the pyramid into four triangular faces and one square base, right? Can you calculate the area of any of those smaller, simpler regions? Given those values, what is the surface area of the whole thing?

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If you're still stuck on any of these, please reply back showing all of your work and reasoning, even if you know it's wrong, and we'll help you further.
 

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