How y The Salid shown in figure consists of a cine of radins (r)and height thit and a cylinderof radins (r) and keight (bz)- The dimensions at any instant It)ares r(t) = 2+t フ プ using the chain rule te find the rate of change fthe Valume of fhe salid
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- StokesTheorem.Evaluate∫ F·dr,whereF=arctanx/yi+ln√x2+y2j+k and C is the boundary of the triangle with vertices (0, 0, 0), (1, 1, 1), and (0, 0, 2).The pressure of a fluid flowing through a closed system, P, after t seconds can be modeled by the fucntion P(t)=100+20sin ((7π/3)t)) In the interval [0,1], determine the times at which the pressure of the fluid is 100mmHg.A force, slows the motion of a weighted spring so that the mass's position at time t is y=22e-t cos(t), t≥0. Find the average value of y over the interval 0≤t≤2π.
- Use the Chain Rule to find ∂z/∂s and ∂z/∂t. (Enter your answer only in terms of s and t. Please use * for multiplication between all factors.) z = x4y9, x = s cos(t), y = s sin(t)Determine if f (x, y) = ={(xsin3y)/(x2+y6), (x, y) ≠ (0, 0) ={1, (x, y) = (0, 0) is continuous at (0, 0)a. Find the center of mass of a thin plate of constant density cov-ering the region between the curve y = 3/x^3/2 and the x-axis from x = 1 to x = 9. b. Find the plate’s center of mass if, instead of being constant, the density is d(x) = x. (Use vertical strips).
- Perform the given problem solving on Finite Divided Difference or Euler's method. A storage tank contains a liquid at depth y where y = 0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Qsin^2(t)İn delta function ; δ(t-t0) =0 , t ≠t0 , integrate δ(t-t0) t=0 to t=infinity equals to 1. According to this what is the Laplace transform of this independent value function? y''+4y=δ(t-π)-δ(t-2π) y(0)=0 y'(0)=0If we think of theguitar string as stretched tight along the x-axis. As the string vibrates, each point onthe string moves back and forth on either side of the x-axis. Let y = f (x, t) = cos t sin x,for 0 ≤ x ≤ π and 0 ≤ t ≤ 2π, be the displacement at time t, in milliseconds, of thepoint on the string located x units from the left end.a. Describe the cross sections of y = f (x, t) when t is fixed.b. Describe the cross sections of y = f (x, t) when x is fixed.c. Explain what the functions f (x, 0) and (x, 1) represent in terms of the vibratingstring.d. Explain what the functions f (0, t) and f (1, t) represent in terms of the vibratingstring. e. Draw the contour diagram for the displacement of the vibrating guitar string. Usethis diagram to describe the behavior of the string.f. Find the rate of change of the displacement with respect to time at a constantposition.g. Find the rate of change of the displacement with respect to position at a constanttime.h. At a fixed time, what point…
- (a) express ∂z/∂u and ∂z/∂y as functions of uand y both by using the Chain Rule and by expressing z directly interms of u and y before differentiating. Then (b) evaluate ∂z/∂u and∂z/∂y at the given point (u, y). z = 4ex ln y, x = ln (u cos y), y = u sin y;(u, y) = (2,π/4)The heat index I is a measure of how hot it feels when the relativehumidity is H (as a percentage) and the actual air temperature is T (in degrees Fahrenheit). An approximate formula for the heat index that is validfor (T, H) near (90, 40) isI(T, H) = 45.33 + 0.6845T + 5.758H - 0.00365T2- 0.1565HT + 0.001HT2(a) Calculate I at (T, H) = (95, 50).(b) Which partial derivative tells us the increase in I per degree increase in T when (T, H) = (95, 50)? Calculate this partial derivative.The volume of a nose cone is generated by rotating the function y = x – 0.2x2 about the x-axis. What is the volume, in m3, of the cone. The volume of a nose cone is generated by rotating the function y = x – 0.2x2 about the x-axis. What is the volume, in m3, of the cone? What is the x coordinate of the centroid of the volume?