math question functions cone URGENT HELP!?

Question

a cone has an altitude of 15 cm and a radius of 3 cm. a right circular cylinder of radius r and height h is inscribed in the cone. write h as a function of f.

Answer 1

In this problem, we need to focus on the half cross-section of the two figures. The half cross-section of the cylinder breaks the half cross-section of the cone into two similar triangles. One triangle sits atop the cylinder. The second triangle is formed by these sides: the height of the cylinder; the part of the half cross-section of the cone which lies between its base and the point where the top of the cylinder touches it, which is actually this triangle's hypotenuse, and last but not least, the radius of the cone minus the radius of the cylinder. A third triangle which is similar to the other two is the half-cross section of the cone itself. We can set up a relationship between the parts of the similar triangles needed to derive the needed function:



This is the particular relationship that obtains in this problem:



height large triangle/height small triangle = base large triangle/base small triangle



Furthermore, it doesn't matter whether we use the triangle atop the cylinder or the triangle at the base of the cone, we get the same relationship. If we use the bottom triangle, we get this:



15/h = 3/(3 - r)

15 (3 - r)/3 = h

(15/3)(3 - r) = h

5 (3 - r) = h or

h = 5 (3 - r)



If we use the top triangle, we get this:



15/15 - h = 3/r

3 (15 - h) = 15r

45 - 3h = 15r

-3h = -45 + 15r

-3h/-3 = (-45 + 15r)/-3

h = 15 - 5r

h = 5 (3 - r)



So the required function is h = 5 (3 - r).

Answer 2

f(x) = x^3 - 4x^2 + 5x - 2 We could have written it this way y = x^3 - 4x^2 + 5x - 2 This simply means that for each value of 'x' there is a corresponding value or values of 'y' I suggest for your own information that you plot the graph of this. This is alway (well usually) very enlightening. When x =0 y = -2 x =1 y = 1-4+5-2 = 0 NOTICE By showing that when x = 1 that y = 0 we have just proved that (x - 1) is a factor. x =-1 y = -1-4-5-2 = -12 x = 2 y = 8 -16 +10 -2 = 0 x = -2 y = -8 -16 -10 -2 = -36 x = 3 y = 27-36 + 15 -2 = 4 x = 4 y = 64-64+20-2 = 18 x =1.5 y = (1.5)^3 -4(1.5)^2 + 5(1.5) -2 = 3.375 -9 + 7.5 - 2 = -0.125 Plot this set of points {-1,-12}, {0,-2}, {1,0}, {1.5, -0.125}, {2, 0}, {3, 4}, {4, 18} Notice the following: The curve rises from the negative area and until it touches the x axis at x = 1, goes negative briefly and turn up again passing through the x axis at x = 2, then it rises steeply into the positive area. We were asked to show that (x - 1) is a factor. We have proved that (x - 1) is a factor when we showed that x = 1 make the function = 0 The usual next step would be to divide this newly found factor into the function f(x) = x^3 -4x^2 + 5x - 2 using long division. ........... x^2 -3x + 2 ..........._______________ (x - 1)| x^3 - 4x^2 + 5x - 2 ............x^3 - x ............_______ ..................- 3x^2 + 5x ..................- 3x^2 + 3x ..................________ ..............................2x - 2 ..............................2x - 2 We see that having divided (x - 1) into x^3 -4x^2 + 5x - 2 we get x^2 -3x +2 We then set about factorising x^2 -3x + 2 Thus we can say x^2 -2x - x + 2 x(x - 2) -1(x - 2) = (x -2)(x - 1) So now the entire factors are (x -1)(x -1)(x - 2) The roots of the equation or zeros of the equation are the values of x at which y is zero and these x values are 1, 1, 2 This means that the factors are (x - 1)(x - 1)(x - 2) Now we must finally look at what remainder we get when we divide the function by (x + 2) Use the same procedure as before: ..........x^2 - 6x + 17 .........______________ (x+2)| x^3 -4x^2 + 5x - 2 .........x^3+2x^2 .........._______ ...............-6x^2+ 5x ...............-6x^2 -12x ...............________ .........................17x - 2 .........................17x +34 .........................______ ...............................- 36 Thus (x^3 - 4x^2 + 5x - 2) divided by (x + 2) = (x^2 -6x + 17) - 36 Is this correct lets check Multiply x^2 - 6x + 17 by (x + 2) and add -36 x^3 - 6x^2 + 17x + 2x^2 - 12x + 34 -36 = x^3 - 4x^2 + 5x + 34 -36 = x^3 -4x^2 + 5x - 2 which is where we started. Thus the remainder 'R' = - 36 I hope you can follow all that expecially the long division.

Answer 3

If I understand the question correctly, the cylinder is contained within the cone. The function requested gives the height of said cylinder given the radius of the cylinder. If this is correct, then the function actually determines the height at which the cone radius equals the cylinder radius.



Assuming a right cone, then a vertical cross section of the cone would be a equilateral triangle with base of 6cm (2 * radius) and a height of 15 cm. If we bisect the triangle, we get a right triangle of base 3 and height 15. The relationship between the height and radius of this right triangle can be determined by the COTANGENT function: cotA = Adjacent (height) / Opposite (radius) = 15 / 3 = 5.



Using the same logic (vertical cross section and then bisecting), the cylinder is represented by a rectangle with width equal to the radius of the cylinder and height to be determined. By overlaying this rectangle within the right triangle with one corner in the right angle of the triangle, the opposite corner will be touching the hypotenuse of the right triangle at height h. This will leave a right triangle above the rectangle that has the same proportions as the original right triangle. We can determine the height of this new right triangle, which we will call h2, using COTANGENT. Then it is a simple matter of subtracting h2 from the 15cm height of the cone to determine the height of the rectangle (i.e. cylinder)



Therefore:



h = h(r) = 15cm - h2

h2 = cot A * r

h2 = 5 * r

h(r) - 15cm - 5 *r



Simplifying:



h= h(r) = 5 * (3 - r)



It would be easier to understand with a diagram, but I don't know how I would do that in this medium.



Hope this helps.

Answer 4

wot is f

Answer 5

I THINK 12cm...