### Problem Statement

My shoelace broke the other day. I was too lazy to take it out and figure out how long it was, so I decided to calculate the length based on the number of eyelets and their separation.

The "horizontal" distance between the eyelets can increase or decrease from the bottom to the top of the shoe. The startWidth is how far apart they are at the bottom (near the toes) and the endWidth is how far apart they are at the top (near the tongue of the shoe). This distance increases (or decreases) linearly from bottom to top. Furthermore, the "vertical" distance, spread, between eyelets is constant.

Write a method, calculate, which calculates the length of the entire shoelace, given the initial distance between the eyelets (startWidth), the final distance between the eyelets (endWidth), distance between each pair of vertically adjacent eyelets (spread) and the number of pairs of eyelets (numPairs).

You should assume all laces follow this general pattern:

```  |----| <- startWidth

O----O    -
\  /     |
/\      |
/  \     |
O    O    -
\  /
\/
/\
/  \
O    O

|----| <- endWidth
```

In this illustration, startWidth and endWidth are identical, and numPairs is 3.

### Definition

 Class: ShoelaceLength Method: calculate Parameters: int, int, int, int Returns: double Method signature: double calculate(int startWidth, int endWidth, int spread, int numPairs) (be sure your method is public)

### Notes

-startWidth may be greater than, equal to, or less than endWidth.
-Your return value must have a relative or absolute error less than 1e-9.
-The units of measurement are irrelevant.
-The length of lace that it takes to go around or through each eyelet should be ignored.
-The length of lace needed to actually tie a knot is not included.

### Constraints

-numPairs is the number of PAIRS of eyelets, and will be between 2 and 50, inclusive.
-startWidth, endWidth, and spread will each be between 1 and 99, inclusive.

### Examples

0)

 `2` `2` `1` `2`
`Returns: 6.47213595499958`
 These are my topsiders. Remember to add the start width across the bottom pair of eyelets.
1)

 `10` `1` `1` `10`
`Returns: 111.1786186482248`
2)

 `1` `10` `1` `10`
`Returns: 102.17861864822481`
 Nearly the same configuration as the previous case, except the width goes from 1 to 10 instead of 10 to 1.
3)

 `1` `1` `1` `2`
`Returns: 3.8284271247461903`
4)

 `1` `99` `19` `2`
`Returns: 107.97663296253066`

#### Problem url:

http://www.topcoder.com/stat?c=problem_statement&pm=2848

#### Problem stats url:

http://www.topcoder.com/tc?module=ProblemDetail&rd=5859&pm=2848

dplass

#### Testers:

PabloGilberto , lbackstrom , brett1479

Geometry, Math