Group+E+Period+8

Welcome Group E Period 8! You must complete problems 5, 11, and 17 in the space provided. Feel free to use the discussion board to bounce ideas off one another. Each member of the group must contribute.

5. Given a rectangular box with lengths x, y and z as the edges of the box (corresponding with length l, width w, and height h in the picture), we must find out
 * how** the derivative of the box's diagonal s (corresponding with diagonal d in the picture), **//ds/dt,// relates to //dx/dt, dy/dt// and //dz/dt//**//.//

The original equation to find diagonal //s// is √(//x//^2 + //y^2 + z^//2) In order to find how //ds/dt// relates to //dx/dt, dy/dt// and //dz/dt//, we must take the derivative of both sides. //s// = √(//x//^2 + //y^2 + z^//2) The derivative of //s// is //ds/dt//. ds/dt = √(//x//^2 + //y^2 + z^//2) Use the power rule to find the derivative of the other side of the equation. //ds/dt// = (1/2)(//x//^2 + //y^2 + z^//2)^-1/2 Next, use chain rule to find the derivative of what's inside the square root. //ds/dt// = (1/2) * (// x //^2 + // y^2 + z^ //2)^(-1/2) * (2x(//dx/dt//) + 2y(//dy/dt//) + 2z(//dz/dt//))

Now, our equation is: //ds/dt =// __2x(//dx/dt//) + 2y(//dy/dt//) + 2z(//dz/dt//)__ 2(√(//x//^2 + //y^2 + z^//2))

The 2's all cancel out, leaving us with our answer, //ds/dt =// __x(//dx/dt//) + y(//dy/dt//) + z(//dz/dt//)__ √(//x//^2 + //y^2 + z^//2)



the balloon is filled with air at a rate of 100(pi) ft^3/sec so (dV/dt)=100(pi) to find the rate the radius increases you take the derivative of the volume formula and set it equal to (dV/dt) and then plug in 5ft for the radius to find the rate of the surface area you take the 2nd derivative of the volume function and plug in 5ft for the radius and 1 for (dr/dt)

17.)

(a) V = (1/3) * pi * r^2 * h r = (15h) / 2

V = (1/3) * pi * (15h / 2)^2 * h = (75 * pi * h^3) / 4 dV/dt = (225 * pi * h^2) * (dh/dt) dh/dt = (4 * -50) / (225 * pi * 25) = -8 / (225 * pi) = -0.0113 m/min = -1.13 cm/min

(b) r = 15h/2 dr/dt = (15/2)(dh/dt) dr/dt = (15/2)(-8/(225 * pi)) = -4 / (15 * pi) = -0.0849 m/sec = -8.49 cm/sec

dV = rate of change in volume dh = rate of change in height dr = rate of change in radius