PROBLEM SOLVING A barn is in the shape of a pentagonal prism with the dimensions shown. The volu

PROBLEM SOLVING A barn is in the shape of a pentagonal prism...

Key Concepts

Composite Solids

Composite solids are shapes that can be broken down into simpler geometric figures whose properties (like volume or surface area) are easier to calculate individually. In problems where a building or structure is made up of different sections, decomposing the overall shape into prisms, pyramids, or other basic shapes is key to setting up equations and solving for unknown dimensions.

Volume of Prisms

This concept involves understanding that the volume of any prism can be found by multiplying the area of its base by its height. It is a fundamental principle in three?dimensional geometry that applies regardless of the specific shape of the base, whether it be a triangle, rectangle, pentagon, or any other polygon.

Right Triangle Geometry and the Pythagorean Theorem

Many roof design problems involve right triangles, where one often needs to find the length of a slope or the height of a triangle given its other dimensions. The Pythagorean theorem is crucial in these scenarios, as it provides a relationship between the legs and the hypotenuse of a right triangle, enabling one to determine missing side lengths when two sides are known.

Algebraic Problem Solving in Geometric Contexts

Many geometric problems require setting up equations based on known formulas and relationships, such as those for volume and the Pythagorean theorem. This approach involves representing the unknown dimensions as variables and then solving the resulting algebraic equations. This blend of algebra and geometry strengthens problem?solving skills and aids in navigating complex, multi-step problems.

Transcript

00:01 Okay, so for this problem, we are looking at a barn, and we are told that it is a pentagonal prism, which i've drawn a rough sketch of the barn here.

00:11 Please notice that it is not drawn to scale, and we are given a few different dimensions.

00:16 So we are given the height of the prism here, which is the length of the barn, because we have to think of the basis of the barn being the actual sided shape.

00:27 So it's a pentagon prism.

00:28 So the pentagons are actually the bases, which we see this surface here and this surface here.

00:36 So the height would be 36.

00:37 We are given two side lengths of 8 and 18 respectively, because this, if you notice, they also draw two right angles.

00:48 So we know this isn't a regular pentagon.

00:50 This is a right, sorry, it's not a regular prism.

00:53 It's a right prism.

00:55 And we're given the volume of the barn, which is 9 ,072 cubic feet.

01:00 Okay so we're asked to find the dimensions of each half of the roof and we are shown that so essentially what we need to solve for is this side here and we know that these are equal because there are the lines drawn through the sides like this in the picture of the actual barn so there is a thought process that needs to take place we need to understand that we don't have the actual area we have the volume so we need to calculate the area that that's a great place to start because what we need to ultimately do is we need to divide this pentagon into two different shapes being this square or not square this quadrilateral this rectangle here with these four right angles and a triangle on the top because this these uh this uh x dimension here would ultimately be the hypotenuse of two right triangles drawn here so we need to be able to divide this base into these different shapes, but to do that, we need to work backwards from the volume first.

02:11 So before we get into that, you know the volume here, and we need to calculate the base area of this pentagon here and here.

02:20 Okay? so the formula for our right prism is volume equals big b times h, where this is the height, and big b is the base area.

02:42 So if we do the math, we just need do the volume divided by the height so we do volume divided by 36 that we see here would be the height let's do the arithmetic so you're welcome to do it if you'd like but i'll do it for sake of the problem so we are given that the base area is 252 and keep in mind that this is the area of one base it's not the total area of both bases because just to find the volume you need to know the area of one base multiplied by the height.

03:24 So 252.

03:25 And this is area now, so we're looking at squared feet.

03:29 So just one of these pentagons, its surface area, is 252 square feet.

03:38 Okay.

03:41 So now we can begin to deduce some more things.

03:47 So remember how i said we need to divide this into shapes.

03:51 That's what we're going to start doing.

03:54 So right away, we know we can calculate the area of this rectangle i just drew which is it's base times height so it's 18 times 8 should we do the math i will do it for everybody right now 18 times 8 is 144 so what that tells us is that out of this 252 144 is the square or the rectangle my apologies so let's just make this a little cleaner.

04:30 So let's do some subtraction.

04:34 Now we know that it's 108 square feet left over that this rectangle is responsible for.

04:45 Okay? and if we draw a line through the middle, we know that it's going to be half the area.

04:51 So let's divide that by two because we're trying to find this hypotenuse.

04:55 That's the best way we can go about this.

05:02 108 divided by two.

05:04 It's going to be 54, which is the area of one triangle so let's just draw that triangle out really quickly one of these right triangles here so the area of this is 54 feet we know that if we made two right angles it needs to be consistent with this line here and the length of the entire big triangle would be 18 if it's split in half evenly then we know that one side will be nine okay so what is the formula for the area of a triangle one half based times height, the base is 9, the area is 54, so it's going to be 54 equals one half of 9 times h.

06:02 So one half of 9 is 4 .5, 54 divided by 4 .5.

06:07 We will do this arithmetic very quickly...

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