Question

Three lines of crops in a garden form a triangular shape. Strawberries are planted in a 10 ft line and green beans are planted in an 18 ft line. If the third side, a line of pumpkins, makes an 68 degree angle with the strawberries, determine the length of the pumpkins.

Answer 1

`p = 19.18`

Explanation:

This question is illustrated using the attachment and will be solved using cosine formula

`a^2 = b^2 + c^2 - 2bcCosA`

Let the strawberry side be s, the Green beans be b and the pumpkins be p.

The cosine formula in this case is:

`g^2 = s^2 + p^2 - 2spCosG`

Where

`s = 10`

`g = 18`

`<G =68`

The equation becomes

`18^2 = 10^2 + p^2 - 2 * 10 * p * Cos\ 68`

`324 = 100 + p^2 - 20 * p * 0.375`

`324 = 100 + p^2 - 7.5p`

Collect Like Terms

`p^2 - 7.5p + 100 - 324 = 0`

`p^2 - 7.5p - 224 = 0`

Using quadratic formula:

`p = (-b\±√(b^2-4ac))/(2a)`

Where

`a = 1`

`b = -7.5`

`c = -224`

`p = (-(-7.5)\±√((-7.5)^2-4*1*-224))/(2*1)`

`p = (7.5\±√(56.25+896))/(2)`

`p = (7.5\±√(952.25))/(2)`

`p = (7.5\± 30.86)/(2)`

`p = (7.5 + 30.86)/(2)` or
`p = (7.5 - 30.86)/(2)`

`p = (38.36)/(2)` or
`p = (-23.36)/(2)`

`p = 19.18` or
`p = -11.68`

But length can not be negative.

So:

`p = 19.18`