Nick has six sticks of the following length: 3 cm, 3cm, 5cm, 9cm, 11cm and 13 cm. how many different triangles can he make using three of these sticks

5 triangles

Explanation:

Think of it this way, a triangle can be formed only if the sum of any two sides of the triangle is greater than the third side, this is called TRIANGLE INEQUALITY.

a + b > c         a + c > b      b + c > a

All three must be satisfied for a triangle to form.

Now given your sticks, there are many possible combinations you could use, but only a few of those combinations will satisfy the inequalities.

I will show you two examples, using the two 3 cm sticks and another stick.

3cm + 3cm = 6 cm

This means that the 3rd side needs to be less than 6cm and there's only 1 option, which is the 5 cm because all the other options are greater than 6 cm.  

TRIANGLE 1 3 cm, 3cm, 5 cm

(This is also the only triangle a 3 cm and 5 cm combination can make)

Next 3 cm and 9 cm combination:

3 cm + 9 cm = 12 cm

It's third side should have a length less than 12 cm.

TRIANGLE 2 3 cm, 9 cm, 11 cm

Again no other side is less than 12 cm.

So you do this for everything else to determine if a triangle is formed.

d. new methods of assembly are introduced.

explanation:

this allows for the goods to be made on a larger scale. not faster because now it is shown to be made better, hence the greater productivity in the question. it is calling for something new to be introduced.