A water tank is in the shape of a right circular cone with height 18 ft and radius 12 ft at the top. If it is filled with water to a depth of 15 ft, find the work done in pumping all of the water over the top of the tank.

Answer:

[tex]y\approx 5\ m[/tex]

Explanation:

The pressure of a Fluid

A fluid of density [tex]\rho[/tex] exerts pressure at a distance y (deep) given by

[tex]P=\rho\cdot y\cdot g[/tex]

Where g is the acceleration of gravity or [tex]g=9.8\ m/s^2[/tex]

This formula computes the pressure assuming the initial pressure is 0 at fluid (water in this case) level.

Knowing the measured pressure, we can know how deep the diver went by solving the equation for y

[tex]\displaystyle y=\frac{P}{\rho\cdot g}[/tex]

Let's plug in the given values

[tex]P=50,000\ Pa= 50,000\ N/m^2[/tex]

[tex]\rho=1,000\ kg/m^3[/tex]

[tex]g=9.8\ m/s^2[/tex]

Thus

[tex]\displaystyle y=\frac{50,000\ N/m^2}{1,000\ kg/m^3\cdot 9.8\ m/s^2}[/tex]

[tex]y\approx 5\ m[/tex]

Answer:

P = 128.6 T / V

Explanation:

The ideal gas equation is

         P V = n R T

Where the pressure is

P = 36 pounds / in²

V = 2500 in³

T = 700 K

    PV = k T

    k = PV / T

    k = 36 2500/700

    k = 128.6

    P = 128.6 T / V

Answer:

Check attachment for complete question

Question

Find a unit vector in the direction in which

f increases most rapidly at P and give the rate of change of f

in that direction; Find a unit vector in the direction in which f

decreases most rapidly at P and give the rate of change of f in

that direction.

f (x, y, z) = x²z e^y + xz²; P(1, ln 2, 2).

Explanation:

The function, z = f(x, y,z), increases most rapidly at (a, b,c) in the

direction of the gradient and decreases

most rapidly in the opposite direction

Given that

F=x²ze^y+xz² at P(1, In2, 2)

1. F increases most rapidly in the positive direction of ∇f

∇f= df/dx i + df/dy j +df/dz k

∇f=(2xze^y+z²)i + (x²ze^y) j + (x²e^y + 2xz)k

At the point P(1, In2, 2)

Then,

∇f= (2×1×2×e^In2+2²)i +(1²×2×e^In2)j +(1²e^In2+2×1×2)

∇f=12i + 4j + 6k

Then, unit vector

V= ∇f/|∇f|

Then, |∇f|= √ 12²+4²+6²

|∇f|= 14

Then,

Unit vector

V=(12i+4j+6k)/14

V=6/7 i + 2/7 j + 3/7 k

This is the increasing unit vector

The rate of change of f at point P is.

|∇f|= √ 12²+4²+6²

|∇f|= 14

2. F increases most rapidly in the positive direction of -∇f

∇f=- (df/dx i + df/dy j +df/dz k)

∇f=-(2xze^y+z²)i - (x²ze^y) j - (x²e^y + 2xz)k

At the point P(1, In2, 2)

Then,

∇f= -(2×1×2×e^In2+2²)i -(1²×2×e^In2)j -(1²e^In2+2×1×2)

∇f=-12i -4j - 6k

Then, unit vector

V= -∇f/|∇f|

Then, |∇f|= √ 12²+4²+6²

|∇f|= 14

Then,

Unit vector

V=-(12i+4j+6k)/14

V= - 6/7 i - 2/7 j - 3/7 k

This is the increasing unit vector

The rate of change of f at point P is.

|∇f|= √ 12²+4²+6²

|∇f|= 14

There's a part of the question missing and it is:

f(x, y) = 4{x(^3)}{y^(2)} ; P(-1,1)

Answer:

A) Unit vector = 4(3i - 2j)/ (√13)

B) The rate of change;

|Δf(1, - 1)|= 4/(√13)

Explanation:

First of all, f increases rapidly in the positive direction of Δf(x, y)

Now;

[differentiation of the x item alone] to get;

fx(x, y) = 12{x(^2)}{y^(2)}

So at (1,-1), fx(x, y) = 12

Similarly, [differentiation of the y item alone] to get; fy(x, y) =

8{x(^3)}{y}

At (1,-1), fy(x, y) = - 8

Therefore, Δf(1, - 1) = 12i - 8j

Simplifying this, vector along gradient = 4(3i - 2j)

Unit vector = 4(3i - 2j)/ (√(3^2) + (-2^2) = 4(3i - 2j)/ (√13)

Therefore, the rate of change;

|Δf(1, - 1)|= 4/(√13)

The wavelength of green light in meters using exponential notation is 5.5 × 10-7 m.

The green light in the visible portion of the electromagnetic radiation spectrum has a wavelength of around 550 nm (nanometers).

To express this wavelength in meters using exponential notation, we can convert nanometers to meters by dividing by 109. So, the wavelength of green light is 5.5 × 10-7 m (meters).

Answer:

Option D is correct.

Explanation:

Bmax = Emax / c

The general form for electromagnetic wave equation is

E = jEmax ×cos(kx-wt)

We were given

(360V/m) sin[ (6.00×1015rad/s)t + (1.96×107rad/m)x ].

So from the equation above

Emax = 360V/m

Bmax = 360/(3×10⁸) = 1.2 ×10‐⁶ T.

Answer

Option D

Amplitude of Magnetic field = B = 1.2×10⁻⁶ T

Explanation:

The relationship between electric field and magnetic field of an electromagnetic wave is given by

B = E/c

Where B is the amplitude of magnetic field and E is the amplitude of electric field and c is the speed of light

The amplitude of electric field is given as 360 V/m

B = (360 V/m)/(3×10⁸ m/s)

B = 1.2×10⁻⁶ V.s/m²

Since 1 Tesla is equal to 1 V.s/m²

B = 1.2×10⁻⁶ T

Therefore, option D is correct

Answer:

the magnitude and direction of the magnetic field in the region through which the current passes is 0.008 T and +z direction.

Explanation:

given information:

current, I = 20 A

magnetic force per unit length, F/L = 0.16 N/m

the conductor in the negative y-direction

θ = 90° (perpendicular)

as we know the formula to calculate magnetic force is

F = B I L sin θ

B = F/(I L sin θ)

   = (F/L) (1/I sin θ)

   = 0.16 (1/15 sin 90)

   = 0.008 T

since F is in the negative y direction, based of the right hand rule the magnetic field is in positive z direction

Answer:

Explanation:

Given:

current, I = 20 A

Magnetic force per unit length, F/L

= 0.16 N/m

Conductor in the negative y-direction, therefore θ = 90° (perpendicular)

For a magnetic field,

F = B I L sin θ

B = F/(I L sin θ)

= 0.16 × (1/15 sin 90)

= 0.008 T

The field is in the +ve z - direction.