29-30 - Equations of Spheres Show that the equation repre- sents a sphere, and find its center and radius. 29. x + y? + z² – 2x – 6y + 4z = 2 30. x + y? + z² = 4y + 4z
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- Find an equation of a sphere with radius r and center C(h, k, l). SOLUTION By definition, a sphere is the set of all points P(x, y, z) whose distance from C is r. (See the figure.) Thus, in terms of r, P is on the sphere if and only if |PC| = . Squaring both sides, in terms of radius r, we have |PC|2 = . In terms of Cartesian coordinates and radius r, the equation of a sphere is (x − )^2+ (y − k)^2+ (z − )^2 =The design of boats is based on Archimedes' Principle, which states that the buoyant force on an object in water is equal to the weight of the water displaced. Suppose you want to build a sailboat whose hull is parabolic with cross-section y=ax^2, where a is a constant. Your boat will have length L and its maximum draft (the maximum vertical depth of any point of the boat beneath the waterline) will be H. See the figure below. Every cubic meter of water weighs 10^4 newtons. What is the maximum possible weight for your boat and cargo?1. A triangle has vertices (1, 1), (3, 7) and (5, 4). What is the distance between the centroid and the line 2x = -y - 2? Hint: The coordinate of the centroid can be obtained from the average of the x and y-coordinates of the vertices. 2. Find the equation of a line passing through (4, -5) when the intercepts are numerically equal but of opposite signs. 3. Find the point (x,4) which is 5 units from the line 12x + 5y - 6 = 0
- 5. Show that the equation represents a sphere, and find its center and radius:x^2+y^2+z^2-8x-10y-2z-39=0I had a question regarding how to find the point on a sphere nearest to other points. I was given this formula x^2 + (y-3)^2 + (z+5)^2 = 4 and asked to find the point nearest to the x-y plane for one part and nearest to the point (0,7,-5) if I could get some insight onto how this problem can be done that'd be great as I'm confused!6. Find an equation of the sphere that passes through the origin and whose center is (3, 1, 3).
- as engineer design a cooling tower in the shape of hyperboloid of one sheet. The horizontal cross sections of the cooling tower are circular with 10m.The cooling tower is 40m tall with maximum cross-sectional radius of 15m. (A) Construct a mathematical equations for this cooling tower. (B) If x=a cos(u) cosh(v) ,y=b sin(u) cosh(v) and z= csinh(v), show that (x,y,z)lies on your equation in Q1(A). (C) A colleague at the same institution want to construct the cooling tower using an hyperbolic cylinder, give reasons for your result in Q1(A) as the best model for the design of cooling tower.13.1.8 With the attached image, please help me find the standard equation of the sphere.Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be the cable connecting Centerville to both towns. The idea is to save on the cost of the cable by arranging the cable in a Y-shaped configuration.Centerville is located at (7,0) in the xy-plane, Springfield is at (0,5), and Shelbyville is at (0,−5). The cable runs from Centerville to some point (x,0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. Find the location (x,0) that will minimize the amount of cable between the 3 towns and compute the amount of cable needed. Justify your answer. 1). To solve this problem we need to minimize the following function of x:f(x)=? a). We find that f(x) has a critical number at x=? b). To verify that f(x) has a minimum at this critical number we compute the second derivative f′′(x) and find that its value at the critical number is =? (positive…
- Centerville is the headquarters of Greedy Cablevision Inc. The cable company is about to expand service to two nearby towns, Springfield and Shelbyville. There needs to be cable connecting Centerville to both towns. The idea is to save on the cost of cable by arranging the cable in a Y-shaped configuration. Centerville is located at (15,0) in the xy-plane, Springfield is at (0,3), and Shelbyville is at (0,-3). The cable runs from Centerville to some point (x,0) on the x-axis where it splits into two branches going to Springfield and Shelbyville. (Draw a picture of this situation).It costs $1000 per unit to lay cable along the x-axis and $1250 per unit otherwise. Write a function for the total cost of laying the cable in terms of x.Cost(x)=__________ Take the derivative of your cost function by carefully applying the chain rule.dCost/dx=__________ To find the x location that yields a minimum cost we need Calculus! Recall that if the derivative of the cost function is zero then the x is…q1.33 you know that a sphere has the equation of x^2 + y^2 +z^2 -6x +4z =3 are the following points inside, outside, or on the sphere? a) (1,1,1) b) (2,2,4) c) (3,4,-2) show all your work clearly.
