#1
+10 What is a counting number? That is whole digits or integers. no fractions.
List all of the numbers from 1 to 59 (the problem does say "numbers less than 60", which I would assume doesn't include 60.)
1,2,3,4,5, ... 59.
Now how many numbers have at least one 5 in it? 5,15,25...
And don't forget that 50, 51, 52, ... 59 all have the digit "5" in it too.
55 is counted as a number that contains a '5', but don't count the two fives, as the number 55 contains (at least one) 5.
I will leave the counting up to you.
Hope that helps.
#2+10 a. In the physical sciences, power means the rate of work done, or work/unit of time.
b. it can take many forms as one reply was suggesting.
Power = Work/time = Force* Distance/ time = Change in Force* Change in distance/ change in time
In calculus form: d(F*d)/dt.
(one can apply the chain rule to the top term.)
In vector Calculus: d(F.d)/dt where the '.' is the dot product.
In other words, its the force in the same direction as the movement that is the power. The perpendicular force doesn't do work ( in a linear universe).
For example; work done lifting an object straight up is m*g*Distance. (m = mass, g = gravitational constant; and the product m*g if the force). Then the power would be: d(m*g*Distance)/d(time).
for uniform gravity and constant mass, m*g*(d[Distance]/d[time]).
Also consider a satellite in circular orbit around the earth. The gravitational force is constant (and a bit lower than at the earth's surface) Assume also the mass doesn't change. then the force is als; F = m*g. However, the motion is perpendicular to the force. So, the dot product between the two vectors is zero. so the work done is zero. That way the satellite has uniform circular motion.
This can also be applied to torque at a lever arm.
Electrical power as well.
In this case the voltage is a potential to do work. the actual work is the amount of charge that is moved through electric field. The formulation is slightly different because the units are normalized to unit charge.
Power = Voltage* Current. (Potential to do work on a unit of chart (V) * Charge flowing in a given time)
in other words: V*Q/t.
Here vector forms can be taken as well. V.dQ/dt, where the dot product is between the electro-static potential and the motion of the charge. As we learned in elementary physics, a charge moving in a magnetic field will be circular. Just like a satellite moving around the earth, the force is perpendicular to earth, a charged particle moving in a uniform magnetic field does no work to speed up or slow down it's motion; only change it direction. So, if not friction occurs on the charge as it moves the path would be circular with uniform speed.
In a circuit, it is assumed that the charge is moving parallel to the electric field and the equations reduce to a simple scaler form; V*I.
Hope that helps because this is very lengthy for a simple question.
