Math Is Fun Forum

It is very simple. You have to use the "strict' triangle inequality which states that the sum of the lengths of any two sides of a triangle is greater then the length of the other side or more precisely if x, y and z are the lengths of the sides of a given triangle then:

Therefore the solutions are:
1-B
2-A
3-D
I hope that answers your question.

It turns out that your approach to this problem related to the "conservation of energy" is incorrect. When calculate the work done by the weight in stretching the elastic string, you have supposed that the force acting on the particle is constant and equal to the weight. However, the tension in the string exerts a force on the weight equal to the sum of the vertical components of the tensions in the AC and CB parts of the elastic string, so the force which exerts a work on the particle is, in fact, variable! Taking this observation into our account, the work done to stretch the string is given by:

where we have used trigonometric substitution and observing that:

and hence

evaluating these integrations and make the substitution:

we would get the result:

This work must equal the elastic potential energy by the conservation of energy. So calculating the potential energies

and

stored in the AC and BC parts of the string as a result of the work done by the force F, respectively, we would get:

Therefore we have:

where k is a constant. Notice that we have treated the modulus of elasticity as a "constant". However, the particle could have any weight!! which is a contradiction. This leads to the conclusion that the modulus of elasticity depends in fact on the force applied to the string and since the force F is variable the modulus of elasticity is also variable. This should be taken into account in the previous calculation which makes it much more sophisticated. Therefore, the second argument which based on the Hook's law is much more simple than the conservation of energy mothed. I hope that will answer your question.

It is absolutely impossible assumption since the gravitational field is conservative hence the work done to reach two points of it is independent of the path between them (in fact, this explains why they did not mention the path taken by the particle in the problem). However, your solution is correct in the sense which I have explained to you, since the problem has no reference to the unit used to measure the weight (in this case they should say something like " a particle of weight W newtons"), so if you reject that reasoning then you are equivalently assuming that the weight is measured by the common units to which there is no reference in the problem. I have reviewed your solution and did not find any mistake even the elastic potential energy is derived correctly from Hook's law:

If your assumption is correct (i.e. the weight should be measured using the common units) then maybe there is something which I didn't notice.

It can fit to the 9 graders, just change the "total surface area" to the "lateral surface area" and  everything will be fine. In fact it is a common practice in mathematics to refer to the lateral surface area as the "surface area" without any restriction. If the problem composer want the reader to include the areas of the bases they usually refer to that by the phrase "total surface area".

It is very simple! the surface areas of the first and the second cones are given, respectively, by:

and since from the givens, we have:

Therefore we get:

so the ratio is 1. I hope that answer your question.

Hi kappa_am, sorry for delay. In fact sciences are human inventions, so we can modify, change or create any concepts to be suitable to answer our needs (always keep in your mind that you can solve any problem, but you have to figure out how to do it). The derivations are very simple. Suppose that we have an oblique triangle with sides a, b and c which are opposite to the angles

and

, respectively, and suppose further that we have taken the angle

to be opposite to the hypotenuse (c in this case), then from the sines law we have:

Now define the generalized Trigonometric functions (suggested by their traditional definitions) as follows:

But from the sines law we have:

and hence:

Which are the required results.

Note: The fundamental identity can be generalized using the cosine law and the other identities have also their generalized versions, but they are very sophisticated (ugly). Please do not hesitate to ask, if you have any question. It was a great time to meet you.

My approach to this problem is to generalize the definition of trigonometric functions by taking the angle

opposite to the hypotenuse to be

. In this case, by using the sine laws, it can be show that:

and hence in your situation the decomposition will be:

Notice that:

The above formulae can be used with any oblique Cartesian coordinate system. I hope that will answer your question.

It can be shown that this probability problem follows the binomial distribution which is given by:

where P, p, N, n are the probability of the n successes out of a total of N for an event of probability p. Therefore, since p=0.5 (because the coin is fair), N=40 and n=10 we have:

Note: If you would like the proof that this type of discrete probability problem does, in fact, follow the binomial distribution, I can provide it to you.

Excellent!!

Hi bob bundy, in fact, your reasoning is correct, since the problem is a weighted arithmetic mean and each probability is actually the weight of the corresponding team (the total of the weights of all terms is 1). You don't have to derive a closed formula of the sum (I don't try, but from my experience, in these types of expressions, there is usually no closed form). However, we have:

therefore, the series converges and hence it has a sum, but if we notice that:

and the remaining terms will be continuously lesser and lesser than that, so the convergence is fast and then we can calculate the average to the accuracy of the second decimal place by simply adding the first seven terms:

I assume that you mean:

If so, the argument goes as follows:

The use of "i" instead of "k" in the last step is valid since "i" and "k" are dummy variables. I hope that will answer your question.

Dear Davidtrinh and Mathegocart, the question simply has something wrong with it. The numbers “30 years” and “94.3 years” are irrational (it is like calculating the length of a person, according to some data, and finding it to be 3 km!!). There is no “bank” that accepts this contract. Actually, in normal circumstances, the bank increases the period of payment if the costumer borrows more money until the period reaches a particular number of years (usually 6-8 years), then they would not increase the period no matter what is the amount that was borrowed. Therefore the problem is unrealistic which is not expected from a textbook. The number “30 years” is probably a typo and the correct number seems to be “3 years”. On the other hand, the formula used by “Mathegocart” is probably not the correct one, since this equation is derived to calculate the saving account balance after a period of time in the case that an amount “P” is deposited in the bank (calculating the monthly payment of a loan in this way is a little bit strange). Finally, even if we assume that the number of years was not “30 years”, there would be a number of possible ways to calculate the period depending on the method applied for calculation. Therefore, without any further information, your question can not be solved “correctly”. This is, in fact, the reason why the brilliant mathematicians in this site did not answer this question.