On a fictional island there are 10 inhabitants, who all know each other, of which 5 are knights, who always tell the truth and the rest of them are knaves, who always lie.

A visitor to the island wants to determine the 5 knaves. What is the minimum number of yes-no questions he must ask the inhabitants in order to find the 5 knaves?

Knights always tell the truth, knaves always lie. Ask them a fact question. => Do you live on an Island?

Knights will always tell the truth, hence, they will all say yes.

Knaves always lie, hence, they will all say no.

The minimum number of yes-no questions that he must ask the inhabitants is 9. (5-yes 4-no/ 5-no 4-yes) At 9, there's enough information to find out the 5 naves.

]]>identical with that in (351 - r ) for r ≧ 7 . E.g. the values obtained in

(173) and (178) are both 239 . The distinct point is that in (173) the

product of 182 is greater while in (178) the product of 165 is greater .

Similarly we can predict that the value obtained in (7) , i.e.

182 * 1 - 165 * 1 = 17 will occur in ( 351 - 7 ) = ( 344 ) .

Let the expression in (344) be expressed as 165 * j - 182 * i = 17

where i is relatively prime with 165 * j while j is relatively prime

with 182 * i .

Since 165 j - 182 i = 17 ⇒ 165 j - 182i = 182 - 165

⇒ 165 j + 165 = 182 i + 182

⇒ 165 ( j + 1) = 182 ( i + 1)

A solution for the equation is j+1 = i+1 = 0 ⇒ j = -1 and i = -1 ,

i.e. the expression in (7) : 182 * 1 - 165 * 1 = 17 .

Another solution is i + 1 = 165 and j + 1 = 182 ,

⇒ i = 164 and j = 181 .

Thus the expression in (344) may be expressed as

165 * 181 - 182 * 164 = 29865 - 29848 = 17

A third solution for the equation may be i + 1 = 2 * 165 = 330

and j+1 = 2 * 182 = 364 ⇒ i = 329 and j = 363 .

According to your earlier comment, you substituted x=y+1 ,expanded and got 3 simultaneous equations in a,b,c. It is o.k. when you are not writing exam or if you are allowed to use programmable calculators but it is awfully simple if you can solve them one by one.]]>

Let's use an example. Suppose that I can choose how many hours I work and that I get paid £9 per hour. Then if H is the number of hours and P is how much I get paid:

P = 9H

Here P depends on H so we call P the dependent variable. I can choose H independently, so H is the independent variable.

Even if the calculation is reversed; how many hours did I work if the pay was £99? ; the names for the variables remain because P is still the one that depends on the other. So writing H = P/9 doesn't alter the names.

Sometimes it isn't clear which is which. eg. There may be a formula connecting 'popularity of a political party' and the 'rate of inflation'. Which is 'cause' and which 'effect'. Who knows? So you cannot always say which is which.

In your example, t is the independent variable because we are choosing values of t; and C is the dependent one because we are using 'f' to work out C depending on what value of t we choose.

Bob

]]>]]>Hi. My snail speed math progression has carried me to a bunch of word exercises in the precalculus book that i'm trying to go through for most of this year, and i am pitifully terrible at word problems:

The box for the new Sasquatch-themed cereal, ‘Crypt-Os’, is to have a volume of 140 cubic inches. For aesthetic reasons, the height of the box needs to be 1.62 times the width of the base of the box. Find the dimensions of the box which will minimize the surface area of the box. What is the minimum surface area? Round your answers to two decimal places.

So my attempt at solving this is

1) find height function (i don't know another way of finding this without making width and length equal):

h = 1.62x

x^2 * 1.62x = 140

h = 140/1.62x^2

2) find surface area function and plug in the height function:

2x^2 + 4xh

2x^2 + 4x(140/1.62x^2)

2x^2 + 560x/1.62x^2

3) I feel like i should find the slant of this function, then find the vertex of this slant, which will give me x, then i should plug that number in surface area function, and get the minimal surface area. Of course none of this is working out for me, so could anyone lend me a helping hand here?

2x < 10 + 5

2x < 15

x < 15/2

x < 7.5

x/3 > 6

x > 6(3)

x > 18

Glad it worked out.

]]>Understanding percentile by an example of the data of marks of students in a class.

Percentile of Score:

To calculate percentile of a student having unique marks in the class:

Percentage of "Students having marks lower than him" over "Total students in class"

To calculate percentile of a student not having unique marks in the class:

Percentage of "sum of students having lower marks than him, and half of all students getting same marks as him" over "Total students in class"

Above we calculated the percentile of a 'student' or better, student's marks.

What if someone asks what is 50% percentile for the class i.e. Score at percentile?

Multiply the percentile number (as a percentage) with the students in the class. Say, there are 40 students in the class, you take 50% of 40 i.e. 20.

If after multiplication the product is a whole number:

then take average of the whole number & the item next to it(data should be arranged in ascending order).

i.e. average of 20th and 21st student's marks.

If after multiplication the product is not a whole number:

then roundup the product, and this item will be the required scoreatpercentile

say, if there were 45 students in the class, then score at 50th percentile will be 23rd item's marks(i.e roundup of 22.5 = 50% of 45)

v = u + at

v is the final velocity; u the initial velocity; a the acceleration and t the time taken.

If an object is thrown upwards the student has to decide if 'up' is the positive direction for velocity, in which case the acceleration is negative. Both are vector quantities. Or you can take acceleration as positive, which makes v negative. It is not usual to consider time as negative, but it is a scalar anyway. I have encountered problems where the solution was a negative t, meaning an event happened before the start time.

Bob

]]>Once of the primary advantages of this is that the primorial can be made once and re-used. This means it's a little trickier for small inputs since the included primes can exceed the input, but we could just use a different method for those.

]]>There's plenty here:

http://www.mathsisfun.com/algebra/quadr … ation.html

But please make your own words and diagrams rather than copying directly.

Bob

]]>The angle of the LED isn't the critical factor. In this diagram I've shown two rays escaping from the sides of the mirror. Ideally, you want all light from the LED to hit the reflector so you get maximum light sent forward.

Bob

]]>One question: in order to use MVT again on f' to get an expression for f", what value do I use for f'(1/2)? Do I use the f'(c) value that I found? [setting up MVT using f'(0) and f'(1/2)?]

Thanks!

]]>