A15 foot ladder is leaning against a wall of a house. the base of the ladder is pulled away from the wall at a rate of 3 ft/sec. find the rate at which the area of the triangle is changing when the base of the ladder is 9 ft from the wall.

7.875 ft/s

Explanation:

L = 15 ft

dx/dt = 3 ft/s

x = 9 ft

Let the top of ladder is coming down with the rate of dy/dt.

use Pythagorean theorem

L^2 = x^2 + y^2     (1)

Differentiate both sides with respect to t

0 = 2 x dx/dt + 2y dy/dt

x dx/dt = - y dy/dt

When, x = 9 ft then y = ? . Put this in equation (1)

15^2 = 9^2 + y^2

225 - 81 = y^2

y  = 12 ft

So

dy/dt = - x (dx/dt) / y = - 9 (3) / 12 = - 9/4 ft/s

Let A be the area of the triangle

A = 1/2 (base)(height)

A = 1/2 (x y)

Differentiate both sides with respect to t

dA/dt = 0.5 (y dx/dt + x dy/dt)

dA/dt = 0.5 [ 12 (3) - 9 (9/4)]

dA/dt = 0.5(36 - 81 /4) = 31.5 / 4 = 7.875 ft/s

a solenoid is created by wrapping a l = 90 m long wire around a hollow tube of diameter d = 4.5 cm.   the wire diameter is d = 0.9 mm.   the solenoid wire is then connected to a power supply so that a current of i = 9 a flows through the wire.

randomized variablesl = 90 m

d = 4.5 cm

d = 0.9 mm

i = 9 a                                          

write an expression for the number of turns, n, in the solenoid. you do not need to take into account the diameter of the wire in this calculation.

calculate the number of turns, n, in the solenoid.

write an expression for the length of the solenoid (l2) in terms of the diameter of the hollow tube d.

calculate the length of the solenoid (l2)   in meters.

calculate the magnitude of the magnetic field at the center of the solenoid in teslas.

explanation: