find num. of rectangles, num. of outfits, min. num. of bks.,

Can someone tell me how to do these problems? (I don't need the answers, I just want to know how to do them.)

How many rectangles are in this picture?


From a wardrobe of 10 sweaters, 8 jeans and 4 pairs of shoes, how many different outfits consisting of a sweater, jeans, and a pair of shoes can a student choose?

How many triangles are in this picture?


YOur math club decides to publish a book of sample problems. The initial cost, regardless of the number of books printed, is $750. After the initial charge, each book costs $2 to produce. What is the smallest number of books your club would need to sell at $5.5o per copy to clear at least 1000?

I dont understand the use of the word "Clear" in the last sentence.

And I was just wondering about this question:

|2x – 3| ? 13

Do I need to flip the inequality symbol for the negative? (2x – 3 ? -13 or 2x – 3 ? -13?)

Re: How do I do these problems?

Kets said:

Can someone tell me how to do these problems? (I don't need the answers, I just want to know how to do them.)

How many rectangles are in this picture? <<< You tell us - just remember two smaller rectangles together make a new bigger rectangle.


From a wardrobe of 10 sweaters, 8 jeans and 4 pairs of shoes, how many different outfits consisting of a sweater, jeans, and a pair of shoes can a student choose?

This is a straight forward question - look at the examples in your text book

How many triangles are in this picture? <<< You count and tell us


YOur math club decides to publish a book of sample problems. The initial cost, regardless of the number of books printed, is $750. After the initial charge, each book costs $2 to produce. What is the smallest number of books your club would need to sell at $5.5o per copy to clear at least 1000?

In these contexts, "clear" means "profit" (revenue - cost = profit)

I dont understand the use of the word "Clear" in the last sentence.

And I was just wondering about this question:

|2x – 3| ? 13

It means

(2x-3) ? 13

and

- (2x - 3) ? 13
Do I need to flip the inequality symbol for the negative? (2x – 3 ? -13 or 2x – 3 ? -13?)

Click to expand...

Re: How do I do these problems?

Thanks! But is there some sort of algebraic way to count the rectangles/triangles?

Re: How do I do these problems?

From a wardrobe of 10 sweaters, 8 jeans and 4 pairs of shoes, how many different outfits consisting of a sweater, jeans, and a pair of shoes can a student choose?

Suppose the wardrobe consisted of 1 sweater, 1 jeans and 4 pairs of shoes, how many different outfits would there be?
Think of 3 blanks to fill in. The first blank is sweaters, the second is jeans and the third is pairs of shoes.
__ __ __
1 1 4 . Do you see there are 4 different outfits? What to do with the 1, 1 and 4 to get an answer of 4?

Suppose the wardrobe consisted of 1 sweater, 2 jeans and 4 pairs of shoes, how many different outfits would there be?
1 2 4 . Do you see that with each of the jeans you would have 4 different pairs of shoes for a total of 8 different outfits? What to do with 1, 2 and 4 to get 8?

Suppose the wardrobe consisted of 2 sweaters, 1 jeans and 4 pairs of shoes, how many different outfits would there be? Answer is 8. What would you do with the 1, 2 and 4 to get 8?

Suppose the wardrobe consisted of 2 sweater, 2 jeans and 4 pairs of shoes, how many different outfits would there be? 16. How do you get it?

Now apply this discovered procedure to the original problem.

Re: How do I do these problems?

Can someone tell me how to do these problems? (I don't need the answers, I just want to know how to do them.)

How many triangles are in this picture?

Given any irregular polygon, or n-gon. Draw "n-3" lines from each vertex to "n-3" points on the opposite sides of the n-gon such that no three of the lines intersect at the same point. How many triangles do all the lines divide the interior of the n-gon into? Only singular triangles are to be counted.

Our first n-gon, with (n+2) sides, is a triangle with 3 sides but no internal lines. Therefore, we have but one triangle.
Our second n-gon is a quadrilateral with two internal lines creating 4 individual triangles.
Our third n-gon is a pentagon with five internal linese creating 10 individual triangles.
Our fourth n-gon is a hexagon with7 internal lines creating 19 individual triangles.
What can we learn from this information that would enable us to define the number of triangles within any n-gon? Lets summarize the data derive so far.

n...................1......2......3......4
Sides............3......4......5......6
Triangles.......1......4.....10....19
1st Difference....3......6......9
2nd Difference.......3......3

An expression can be derived enabling the definition the nth term (or the (n+2)-gon) of any finite difference series. The expression is a function of the number of successive differences required to reach the constant difference. If the first differences are constant, the expression is of the first order, i.e., N = an + b. If the second differences are constant, the expression is of the second order, i.e., N(n) = an^2 + bn + c. Similarly, constant third differences derive from N = an^3 + bn^2 + cn + d. Using the data points (n1, N1), (n2,N2), (n3,N3), etc., from our data, we substitute them into N = an^2 + bn + c as follows:

(n1,N1) = (1,3) produces a(1^2) + b(1^1) + c = 3 or a + b + c = 3
(n2,N2) = (2,4) produces a(2^2) + b(2^1) + c = 4 or 4a + 2b + c = 4
(n3,N3) = (3,10) produces a(3^2) + b(3^1) + c = 10 or 9a + 3b + c = 10

Subtracting successive pairs leads us to a = 3/2, b = -3/2, and c = 1. Therefore, our expression for the the number of triangles within the nth n-gon having (n+2) sides is N(n) = (3n^2 - 3n + 2)/2.