What is the radius of the table?

[loli12]

Anyone has any clue to these questions? They are from a contest last year.
Please Help and if you don't mind... please explain in detail.. Thanks!

1. A library has between 1000 and 2000 books. Of these, 25% are fiction, 1/13 are biographies and 1/17 are atlases. How many books are either biographies or atlases?

2. A circular table is pushed into a corner of a rectangular room so that it touches both walls. A point on the edge of the table between the two points of contact is 2 inches from one wall and 9 inches from the other wall. What is the radius of the table?

3. Let f(x) = ax + b, with b<a both positive integers. If for positive integers p and q, f(p) = 18 and f(q) = 39, what is the value of b?

4. In Triangle SBC, SB = 12, BC = 15, and SC=18. Let O be the point for which BO bisects angle SBC and CO bisects angle SCB. IF M and L are on sides SB and SC respectively so that ML is parallel to side BC and contains point O, what is the perimeter of Triangle SML?

 

[Gokul43201]

1. What number between 1000 and 2000 can be divided exactly by 4, 13 and 17 ?

2. Let the co-ordinates of the point be $(rcos\theta, rsin\theta )$. You now have 2 equations in 2 unknowns.

3. Subtract the two equations. You get only 4 possible choices for the values of a. Three of these are ruled out by inspection - in fact, it's plain to see why. The surviving one works.

4. Let SO extended meet BC at R. Since O is the incenter, we have (I'm not sure what this rule is called) SO/OR = (12+18)/15. From here, it's just similar triangles.

 

[tacman]

1. To solve this problem, we need to first find the total number of books in the library. We are given that the library has between 1000 and 2000 books, so the total number of books can be represented as 1000 < x < 2000, where x is the total number of books.

We are also given that 25% of the books are fiction, so the number of fiction books can be represented as 0.25x. Similarly, 1/13 of the books are biographies, so the number of biographies can be represented as (1/13)x. And 1/17 of the books are atlases, so the number of atlases can be represented as (1/17)x.

To find the number of books that are either biographies or atlases, we need to add the number of biographies and atlases together. So the total number of books that are either biographies or atlases can be represented as (1/13 + 1/17)x = (30/221)x.

Now we can substitute x with the given range of 1000 < x < 2000 to get the range of the number of books that are either biographies or atlases.

(30/221) * 1000 < (30/221)x < (30/221) * 2000

So the range of the number of books that are either biographies or atlases is approximately 136 books to 272 books.

2. To find the radius of the table, we can use the Pythagorean Theorem.

Let the radius of the table be represented as r. We are given that a point on the edge of the table between the two points of contact is 2 inches from one wall and 9 inches from the other wall. This forms a right triangle with the radius of the table as the hypotenuse.

Using the Pythagorean Theorem, we can set up the equation:

r^2 = 2^2 + 9^2

Simplifying, we get r^2 = 4 + 81 = 85

Taking the square root of both sides, we get r = √85 inches.

3. To find the value of b, we can substitute the given values into the equation f(x) = ax + b.

We are given that f(p) = 18