Critical Formula #2 :: Laws of Exponents

These laws are important throughout SPM and Pre-Calculus, so mark a note on them.

That is not the Total Area, or specifically, your triangle area of the side of the pyramid is incorrect!

You should thank Flame Haze for showing the detailed workings. Your problem-solving approach is practically right. The triangle Δ area formula is ½×(base of Δ)×(height of Δ), where height of Δ is the slant height of the pyramid.



First, we settle the total surface area of the pyramid:



Applying Pythagoras' theorem to the right triangle of the pyramid, it is VERY OBVIOUS that h² + (b/2)² = s²:



Then, we settle the Volume of the pyramid V = (1/3)(b²h):



Differentiate



To find the maximum volume, set dV/db = 0, and solve for b:



Here is the plot of the volume function, V(x):

You mean Implicit Differentiation? Of course!

Depends on the behavior of function V(x), the graph must not have a global maxima and a local maxima at the same time. Since the graph (Quartic function) is symmetrical, having 2 repeated roots at the origin, and 2 distinct roots (equal magnitude but different sign), therefore, your bold approach is perfectly feasible in this case. From the graph y = V², you can see



Compare it to original function V(x). Beyond x > 6 and x < -6, the function requires complex number treatment.



Your approach does not work in the following graph because it has a global maxima and a local maxima.



This post has been edited by Critical_Fallacy: Jan 1 2014, 07:18 PM

As you probably already know, we don't actually divide matrices. Try this:

OK! This is going to be fun. Projectile motion of an object at STPM/A-level that is simple to analyze if we make two assumptions: (1) the free-fall acceleration is constant over the range of motion and is directed downward, and (2) the effect of air resistance is negligible.

You probably have learned that the horizontal range R of the projectile is given by



Thus, the maximum value of R from the above Equation is:



This result makes sense because the maximum value of sin 2θ is 1, which occurs when θ = 45°.

Imagine a motorcycle rider rides horizontally off the edge of a cliff, launching himself into a parabolic path that ends on the flat ground at the base of the cliff. If he survives and repeats the experiment with increased launch speed, he will land farther from the starting point. We can imagine him launching himself with great enough speed that the earth’s curvature becomes significant. As he falls, the earth curves away beneath him. If he is going fast enough, and if his launch point is high enough that he clears the mountaintops, he may be able to go right on around the earth without ever landing.

Can you ask your physics teacher whether or not, it is possible to estimate the Escape velocity from this range formula? The following illustration is based on one in Isaac Newton’s Principia.



This post has been edited by Critical_Fallacy: Jan 2 2014, 01:27 AM

Given the Instantaneous Acceleration, we can separate the variables. Integrating both sides of the equation to obtain a solution:



From the Instantaneous Velocity, we can derive the final position of a particle:



Rearrange the equation to obtain the displacement:



See the difference?

Ask your A-level Physics teacher. Something to ponder. Assuming the cannon fires at the same speed as the Earth's rotation speed, where does it land?

Hi maximR & RED-HAIR-SHANKS

Physics Think Tank

An inexperienced archer aims and shoots an arrow straight at an acorn that is being held by Scrat on a cliff. At the same instant that the arrow leaves the bow, Scrat drops the acorn. Ignoring air resistance, does the arrow hit the acorn, Scrat, or neither?

Perhaps you can study this example. Please check for any mistake.

Brilliant solution!

I overlooked the word "system" and treated them as isolated equations.

Luckily v1n0d and studyboy are here!

Sorry, I'm bad at discrete math. Can you check this working?

studyboy is right! In other words, both x(s) and y(s) are parametric equations and we have to deal with the system of implicit parametric equations with Implicit Differentiation. Sweet!

You're right, and there is a condition imposed by the initial velocity as mentioned by v1n0d.

If there were no gravity, the arrow would fly straight to Scrat and the acorn. Since gravity gives the dropped acorn and the released arrow the same constant acceleration downward, they each fall the same vertical distance below the positions they would have had with no gravity. Thus, the arrow ends up hitting the acorn no matter what the initial velocity** of the arrow. The higher the velocity of the arrow, the sooner they meet and the shorter the vertical distance that the acorn falls before being hit.



** Assuming the arrow is fired from ground level, the initial velocity of the arrow must be higher than the minimum velocity given by:



where hmax is the height of the acorn from ground before it is dropped, xmin is the horizontal distance of the archer away from Scrat / acorn, and tmax is the time of the acorn strikes the ground after it is released by Scrat.

The question is from kingkingyyk. Perhaps we should let him clarify here. It is the notation for composite function.

Can you put up your answers for Part (a) & Part (b), and check the formulas for electrostatic potential & gravitational potential (look at your textbook) here? It should obey the superposition principle of potentials.

To find the electrostatic potential at C, the superposition principle can be applied:



To find the gravitational potential at C, the superposition principle can be applied:

Thank you for your explanations on Composition of Relations. Appreciate your help on Combinatorics to kingkingyyk.

If I'm not mistaken, you will eventually study Graph Theory in Computer Science, one of the fastest-growing areas.