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- How are the absolute maximum and minimum similar to and different from the local extrema?Suppose you are climbing a hill whose shape is given by the equationz= 1000−0.01x2−0.02y2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (50, 80, 847). The positive x-axis points east and the positive y-axis points north. (a) If you walk northwest, will you start to ascend or descend? At what rate? (b) In what direction is the slope largest? Why?a particle is moving along the curve y = 2sqrt(5x+1). As the particle passes through the point (3,8) its x coordinate increases at a rate of 3 units per second. Find the rate of change of the distance from the particle to the origin at this instant.
- A particle moves on the line y = 2x + 1 in such a way that its x−coordinate changes at a constant rate of 4 units per second. A right triangle is formed by the vertical line from the particle to the x−axis, the line connecting the particle to the origin, and the x−axis. At what rate is the area of the right triangle changing when x = 4?Water is flowing at the rateof 6 m3/min from a reservoir shaped like a hemispherical bowl ofradius 13 m, shown here in profile. Answer the following questions,given that the volume of water in a hemispherical bowl ofradius R is V = (π/3)y2(3R - y) when the water is y meters deep. At what rate is the water level changing when the water is 8 m deep?A rectangle is inscribed in the ellipse 16 x2 + 81 y2 = 1296 Find the rate of change of the area of the rectangle with respect to xRate of Change of Area with respect to x =
- A point is moving along the circle x^2+y^2=r^2 in the first quadrant such that its x-coordinate changes at the rate of 2.0 units per second. How fast is its y-coordinate changing as the point passes through the point where x = 5.2 units. The radius r of the circle is 8 units.solve the problem with the second partial derivative test or with Lagrange's method: Find the volume of a rectangular box with maximum volume if the surcafe is 96cm2 please show full and complete procedure HANDWRITTEN onlySolve for the general solution using integrating factors found by inspection.
- A point is moving along the circle x2 + (y − 2)2 = 26 in such a way that its x-coordinate is decreasing at a rate of 3 unit per second. What is happening to the y-coordinate at the instant when the point has reached (−5, 3)? The y-coordinate is decreasing at a rate of ________ units per second.A metal storage tank with fixed volume V0 = 360π m^3is to be constructed in the shape of a right circular cylinder (including the bottom) surmounted by a hemisphere. A picture is given below.What dimensions will require the least amount of metal? Show and organize your work; use the firstderivative test or the second derivative test or other tools to check that yours is the desired optimalsolution.A spherical snowball is melting in such a way that its radius is decreasing at a rate of 0.1 cm/min. At what rate is the volume of the snowball decreasing when the radius is 8 cm. (Note the answer is a positive number). The altitude of a triangle is increasing at a rate of 22 centimeters/minute while the area of the triangle is increasing at a rate of 55 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 1010 centimeters and the area is 8484 square centimeters?