need help with geometry

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Jun 5th, 2015

Thank you for the opportunity to help you with your question!

We can see that we have a cube and a pyramide. So the total volume would be the volume of the cube plus the volume of the pyramide. So first, we find the volume of the cube by using this formula:

Vcube = x^3 ; where x is the length of the sides of the cube.

Here, we can see that the lenght of the sides of the green cube is 15 ft (x = 15ft). Then we enter it into formula:

Vcube = (15ft)^3 = (15ft)(15ft)(15ft) = 3375 ft^3

Then we need to find the volume of the pyramide by using this expression:

Vpyramide = (Abase)(height)/3 ; Where Abase is the area of the base.

So we can see that the area of the base is a square which its length side is 15 ft. Let's remember

the formula of the area of a square is:

Asquare = x^2 = (15ft)^2 = (15ft)(15ft) = 225 ft^2

Now we need to find the height of the pyramide. Let's notice that we have a right triangle with a hypotenuse of 9 ft and its base would be the half of the length of the square (15ft/2 = 7.5 ft) and its height would be the height of the pyramide. Then we can use Pythagoream theorem:

c^2 = a^2 + b^2 ; c is the hypotenuse, a and b are the other sides of the right triangle (its height and base).

So c = 9 and b = 7.5. Then we solve for a like this:

9^2 = a^2 + 7.5^2  ----------->  a = sqrt(81 - 56.25) = 4.97 ft

Finally, we have:

Vpyramide = (225ft^2)(4.97ft)/3 = 372.75 ft^3

Vtotal = Vpyramide + Vcube = 372.75 + 3375  = 3747.75 ft^3

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jun 5th, 2015

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