Question

100 points

Find the Surface area of a right rectangular pyramid 50 points for explaining in full the equation of how.
L= 7
W= 5
H= 12
100 points explaining the equation for volume too.

Answer 1

Step-by-step explanation for Surface Area:

So, to find the surface area, you use the formula:
`A=lw+l\sqrt{((w)/(2))^2 +h^2 } +w\sqrt{((l)/(2))^2 +h^2 }`

I'll explain what each part means and why they are like that. (btw, it would help you more if you looked at a rectangular prism I added while you read the explanation :) )

First of all, you do lw because that is the base of the right rectangular pyramid (since you're finding the surface area, you need to find the area of all the surfaces on the pyramid)

You add
`l\sqrt{((w)/(2))^2 +h^2 }` because now you're trying to find the triangular side with the length. You multiply the length, because that's how wide the "triangle" is. You do the square root because of the pythagorean theorem (since you're using the hypotenuse times the length to find that side). The width is divided by two, because as you can see in my attachments, that's how far away the height is from the length (if you look at the base, see where the height line drops in comparison to the length). (look at my second screenshot)

You add
`w\sqrt{((l)/(2))^2 +h^2 }` for the same reason, except dealing with the width. You're finding the side with the width. You multiply the width by the hypotenuse of the right triangle (look at the third screenshot i attached). The length is divided by two because that's the distance between the bottom of the height line to the w.

I do believe that this should explain the meaning behind each piece of the formula (I'm not sure this is what you asked for lol but it does make understanding it a little easier, I think). To get the answer, you just substitute each variable with those values. So...

`A=(7)(5)+(7)\sqrt{(((5))/(2))^2 +(12)^2 } +(5)\sqrt{(((7))/(2))^2 +(12)^2 }`

A ≈ 183.3

As for the Volume:

`V=(lwh)/(3)`. I'm not 100% on why it's divided by three, but with any volume, you're always going to end up with an answer that is in cubed units. So that explains the three variables in the numerator. I'd like to think the three just has to do with the three variables, but I don't know. Give me the points if you'd like? I don't think this is a proper explanation though.

Good luck!

Answer 2

volume = 140 cubic units

area = 183.3 square units

Explanation:

I am going to assume we are dealing with a pyramid with a rectangular base. The rectangular base has length 7 units, width 5 units, and height 12 units.

Let's do the volume first since it's simpler.

Volume:

volume of pyramid = (1/3) * (area of base) * height

volume = (1/3) * 7 units * 5 units * 12 units

volume = 140 cubic units

Now we deal with the surface area.

The total surface area of a rectangular pyramid is the sum of the lateral area and the area of the base. The area of the base is the area of a rectangle. The lateral area is the sum of the areas of the 4 triangular sides. Each two opposite sides are congruent. For the base, we have the length and width of the rectangular base, so we can find the area easily.

For the 4 triangular sides, we need to find the the height of the triangles. Since each pair of opposite faces is congruent, we have two triangles we need to find the heights for. I'll call the heights "p" and "q". We use the Pythagorean theorem to find the heights of the triangular faces.

a^2 + b^2 = c^2

(3.5)^2 + (12)^2 = p^2

p^2 = 156.25

p = 12.5

(2.5)^2 + (12)^2 = q^2

q^2 = 150.25

q = 12.25765

Area of triangle = (1/2) * base * height

For two triangles, the base is 5 and the height is 12.5.

For the other two triangles, the base is 7 and the height is 12.25765.

total surface area = area of base + 2(area of triangle with base 5) + 2(area of triangle with base 7)

A = LW + 2(1/2)(base)(height) + 2(1/2)(base)(height)

A = 7(5) + 2(1/2)(5)(12.5) + 2(1/2)(7)(12.25765)

A = 183.3 square units