Math HELP CONE CUBE geometry?

Question

The radius of a cone is 8 and the height is 5. Find the side of the cube inscribed in these cone.

PLEASE HELP I URGENT

Answer 1

I have never done a problem like this one. But I think you mean the largest inscribed cube.

I proceded by thinking the side of the largest cube would be equal to sqrt(2)*radius of the section of the cone where the top of the cube would fit. The sqrt(2) comes from the largest square one can fit in a given circle.

Then you need an expression for the radius as a function of the side of the cube. This linear and related to the slant of the cone. I got this: Radius = 5 - (8/5)*side.

as stated. Radius = side/sqrt(2)

substitute and solve.

I ended up with some crazy expression like:

25*sqrt(2) / (5 + 8*sqrt(2) about 2.17, I think.

What do you think?

Answer 2

Imagine cutting the cone in the vertical direction, through a diagonal of the cube. Then cut it again in the vertical direction, through the leftover vertex. You will have a quarter of a cone and a quarter of a cube. You will see two similar triangles. The big right triangle will have base r = 8, height h = 5, and its hypotenuse will be an edge on the outside of the cone. The smaller right triangle will have an edge of height x, a base 8 - 0.5 x sqrt(2), and a hypotenuse that will be an edge on the outside of the cone. Here x = length of the cube. How did I get the base? The diagonal of the cube = x sqrt(2) since it is the hypotenuse of a 45-45 triangle.
Using similar triangles,
(5/8) = x / ( 8 - 0.5 x sqrt(2) )
5 - x (5/16) sqrt(2) = x
x (1 + (5/16) sqrt(2) ) = 5
x (16 + 5 sqrt(2) ) = 5 (16)
x = 80 / (16 + 5 sqrt (2) )
Answer: x = 4.6 inches

Answer 3

i think of that the least perplexing thank you to try this in elementary terms with the help of geometry is to apply the thought of Pappus-Guldinus. This says that any quantity which would be generated with the help of sweeping a plane shape alongside a course is comparable to the fabricated from the part of the plane shape and the dimensions of the path of the centroid of the plane shape. i will coach you techniques it applies right here then clarify how the thought may well be derived from geometry on my own. contained in terms of a cone, the plane shape is a acceptable triangle of top h and radius r. The centroid, or stability element of a acceptable triangle is located a million/3 the way from the desirable suited attitude to the different vertices. in case you positioned the desirable suited attitude at an beginning place, then the centroid is at (r/3, h/3). To generate a cone, revolve this triangle with regard to the vertical part of length h. this suggests that the path of the centroid is an entire circle of radius r/3 giving a course length of 2?(r/3). the part of the triangle is rh/2. the quantity is the fabricated from those 2: V = 2?(r/3) * rh/2 = ? r^2 h/3 you will discover the quantity of many complicated shapes employing this theorem. the thought works because of definition of the centroid. in case you think of the plane shape following some curving course, that's elementary to coach that, because it plane bends decrease backward and forward, the quantity lost with the help of the section interior the curve precisely fits the quantity further exterior the curve. This assertion is merely authentic if the path observed is often on the centroid of the form. you will discover the centroid of a triangle with the help of the geometric shape of drawing lines from each and each vertex to the midsection of the different section. the two the thought of Pappus-Guldinus and the alternative of the centroid have been executed with the help of geometry long earlier calculus grew to become into invented so the quantity of a cone with the help of geometry on my own is unquestionably documented.

Answer 4

too hard to do for me.