3D Geometry Questions and Answers

This is an essay type question. Although Canvas provides space for you to type an answer, you do not need to type anything. This question will be graded completely based on the work that you submit at the end. A. Find the equation of a sphere passing through the point (-1, 2, 4) with center (2, 5,1)? B. Describe the intersection of this sphere with the xy-plane.

Two pyramids are similar with a ratio of surface areas of 25:64. Find the volume of the second pyramid given that the first has a volume of 250 m³. A 61.0 m³ B 640 m³ C 1024 m³ D 97.7 m³ E 4194 m³

For the equation below, eliminate the xy-term and find the standard form of the conic in terms of x' and y'. Sketch the conic on the rotated axes. 41x²-24xy +34y²-25=0

Let VC R5 be the subspace V span {(1, 0, -3, 5, 0), (0, 2, 0, 0,3)} and let w = (-5, 0,5, -3, -4). (a) Find the orthogonal projection of w onto V, using exact values in your answer. (b) Find the component of w orthogonal to V using exact values.

Assume that you want to approximate a function f(x, y, z)=x² + y² +2² by a line function ((x, y, z)= a₁x+ay+at+e such that f(x, y, z)=(x, y. 2) at four points (1,0,0), (1, 1.0), (0, 1, 1), a (1,0,1). Here, means that f(x, y, z) is almost equal to f(x, y, z). Your goal is to find the coefficients a₁.a. and c. Model this problem into an optimization problem of the form that minimizes the sum of squar errors between f and at these four points. Solve the resulting optimization problem.

(b) What is the group of all symmetries of the tetrahedron, including those that reverse orientation (and are therefore no longer rotations in 3-space)? (Hint: it's S₁. Why?)

The diagram represents a regular pentagonal pyramid. The measurements unit is feet. (Round your answers to two decimal places.) (a) Find the area of the base. ft² (b) Find the lateral area of the pyramid. ft² (c) Find the surface area of the pyramid. ft² (d) Find the volume of the pyramid. ft3

Rewrite the following rectangular equation in polar form. y=3x

Find an equation of the line where the planes Q and R intersect. Q: -2x+3y-z = 1; R: x+y+z= -1 Choose the correct answer below. A. An equation of the line is given by x = -5t, y = -1/5+4t,z=-4/5+t, where -∞<t<∞. B. An equation of the line is given by x =-4/5+4t,y=-1/5+t,z=-5t,where -∞<t<∞. C. An equation of the line is given by x =-5/4+4t,y=-5+3t,z=-2t,where -∞<t<∞o. D. An equation of the line is given by x= -5+3t, y = -2t, z =-5/4-t,where - ∞<t<∞.

Which revolution will generate a cylinder? None of the other answers are correct Rotate a circle 360° about one of its diameters. Rotate a rectangle 360° about one of its diagonals. Rotate a circle 360° about one of its tangent lines. Rotate a rectangle 360° about one of its sides.

Identify the transformations. Sketch a graph of the following functions as a transformation of the graph of a toolkit function by first graphing the toolkit function, then the progression of transformations (in the correct order). f(x) = -3(-x - 5)² + 4

What is the exact volume of the solid stretching over the x-interval [0, 4] whose cross-sections are equilaterial triangles where the length of the side of a cross-section at x is given by the function f(x)=√49-x²? Note: Maintain fractions if applicable.

The density of water is 1 gram per cm^3. An object floats in water if its density is less than water's density, and it sinks if its density is greater than water's. Will a toy in the shape of a rectangular prism that is 1 centimeter by 2 centimeter by 2 centimeter with mass 3 grams sink or float? Explain your reasoning.

Draw the shape, Find (b) period, (c) amplitude, (d) width, (e) shift (f) (x₀), x₁, x₂, x₃, x₄, x₅, and (g) sketch a graph of the function through one full period, . f(x) = – 3 sec(2πx + π/6) + 1.

Which of the following statements is true? Enter a, b, c, d, or e. a. All trapezoids are parallelograms. b. All squares are rectangles. c. All rhombus's are squares. d. Trapezoids are not convex. e. None of the above.

Which of the following statements is false? Enter a, b, c, d, or e. a. All squares are rectangles. b. All rhombuses are squares. c. All trapezoids are polygons. d. All rectangles are quadrilaterals. e. None of the above.