Temperature Let
Use Lagrange multipliers to find the maximum temperature on the curve formed by the intersection of the sphere and the plane
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Calculus (MindTap Course List)
- please do not provide solution inimage format thank you. A thin metal plate located in the center of the xy plane has a temperature T(x, y) at the point(x,y) given by T(x, y) = 100/(1 + x^2 + y^2) . (a) What is the temperature on the plate at point (1, 2), approximately? (b) At what point is the temperature as high as possible? (c) If a particle moves away from the origin, moving along the positive x axis, Will the temperature increase or decrease? (d) At what points is the temperature 50? (e) The contour lines of T are called isotherms (because all points on a of these curves have the same temperature). Sketch some isotherms of that function.arrow_forwardDetermine the location of a point (x, y, z) that satisfies the condition. x > 0 a) The point is behind the yz-plane. b) The point is behind the xz-plane. c) The point is in front of the yz-plane. d)The point above the xy-plane. e)The point below the xy-plane. f)The point is in front of the xz-plane.arrow_forwardUse Lagrange multipliers to find the highest point on the curve of intersection of the surfaces. Cone: x2 + y2 − z2 = 0 Plane : x + 2z = 4arrow_forward
- find a parametrization of the surface. 1.The upper portion cut from the sphere x2 + y2 + z2 = 8 by the plane z =-2 2. Parabolic cylinder between planes The surface cut from the parabolic cylinder z = 4 - y2 by the planes x = 0, x = 2, and z = 0arrow_forwardCompound surface and boundary Begin with the paraboloidz = x2 + y2, for 0 ≤ z ≤ 4, and slice it with the plane y = 0.Let S be the surface that remains for y ≥ 0 (including the planar surface in the xz-plane) (see figure). Let C be the semicircle and line segment that bound the cap of S in the plane z = 4 with counterclockwiseorientation. Let F = ⟨2z + y, 2x + z, 2y + x⟩.a. Describe the direction of the vectors normal to the surface thatare consistent with the orientation of C.b. Evaluate ∫∫S (∇ x F) ⋅ n dS.c. Evaluate ∮C F ⋅ dr and check for agreement with part (b).arrow_forwardCylinder and sphere Consider the sphere x2 + y2 + z2 = 4 andthe cylinder (x - 1)2 + y2 = 1, for z ≥ 0. Find the surface areaof the cylinder inside the sphere.arrow_forward
- A bowl has inner surface given by the graph of the function z = f(x, y) = 2x^2 + 3y^2. A drop of oil is placed on this surface at the point (2, 1, 11) and moves along the surface under the influence of gravity toward the point (0, 0, 0), with position function (x(t), y(t), z(t)). The projection into the xy-plane of its position is the pair (x(t),y(t)). Assume that gravity causes the drop to move so that the projection moves in the direction of the negative of the gradient vector of f. Find the curve in the xy-plane above which the drop moves. Give your answer in the form y = some function of x. this is all the information we were givenarrow_forwardHow do I find the point on a sphere x2 + (y - 3)2 + (z + 5)2 = 4 nearest to a.) the xy-plane b.) the point (0, 7, -5).arrow_forwardFind equations for all the planes that intersect the y-axis at y = 1 and the z-axis at z = 2, and are tangent to the sphere (x-2)^2 + y^2 + z^2 = 4. Do not use calculusarrow_forward
- Volume of the solid when R is revolved about the y-axis Y=x Y=7x Y=28arrow_forwardy² - (z²/3) = 1 + (x²/12) sketch the quadric surface. emphasize the traces on the coordinate planes and the plane y = +/- 2.arrow_forward(a) Find the overlapping area of two equations also give its geometrical representation: u^2+v^2=n, u^2+(v-√n)^2=1 Where n is your arid number for example if your arid number is 19-arid-1234 then n=1234. (b) Find the points that touch the x axis of curve v=u^2+pu+(n+1). Where n is your arid number for example if your arid number is 19-arid-1234 then n=1234. Also find the equation of lines at that points give geometrical representationarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage