Work done by friction on an incline plane

[idllotsaroms]

Homework Statement

http://postimage.org/image/gprwhko37/

Could someone check my work to make sure I'm doing this problem correctly?
Where:
s : hypotenuse or distance
h : height
uk : coef friction

Homework Equations

W = F*D
Force Normal = mgcosθ
Wfriction = (uk * N) * s(distance)

The Attempt at a Solution

Solve for θ to use to find force normal
Sinθ = h / s
θ = arcsin(h/s)

Y component of weight of box = mgcos*arcsin(h/s)
Fy = may = N - mgcos*arcsin(h/s)
N = mgcos*arcsin(h/s)

Then plug N into equation Wfriction = -uk * N * s
Wfriction = -uk*mgcos*arcsin(h/s)*s

 

[rude man]

After a few extraneous equations, the result looks right.

 

[idllotsaroms]

After a few extraneous equations, the result looks right.

Thank you Rude man, but how would I remove the extraneous equations? Could you show me an alternate shorter way to go about doing this?

Also, would I have to solve for a when using Fy = may = N - mgcos*arcsin(h/s)?

 

[rude man]

Homework Statement

http://postimage.org/image/gprwhko37/

Could someone check my work to make sure I'm doing this problem correctly?
Where:
s : hypotenuse or distance
h : height
uk : coef friction

Homework Equations

W = F*D
Force Normal = mgcosθ
Wfriction = (uk * N) * s(distance)

The Attempt at a Solution

Solve for θ to use to find force normal
Sinθ = h / s
θ = arcsin(h/s)

Y component of weight of box = mgcos*arcsin(h/s)
Fy = may = N - mgcos*arcsin(h/s)
Omit the red.

N = mgcos*arcsin(h/s)
Then plug N into equation Wfriction = -uk * N * s
Wfriction = -uk*mgcos*arcsin(h/s)*s

see above.