High School

Compute the work done by the force field [tex]F(x,y)=\langle5x,5y\rangle[/tex] along the curve [tex]C[/tex], where [tex]C[/tex] is the line segment from [tex](2,1)[/tex] to [tex](3,5)[/tex].

A) \(\frac{145}{2}\)

B) 25

C) 145

D) \(\frac{2}{25}\)

Answer :

The work done by the force field F along the curve C, given the force field, is C. 145.

How to find the work done ?

In this case, the force field is F(x, y) = ⟨5x, 5y⟩ and the curve C is the line segment from (2, 1) to (3, 5).

The vector element along the curve can be written as:

dr = (dx, dy)

Substituting the force field and vector element into the line integral, we get:

∫CF · dr = ∫(5x, 5y) · (dx, dy)

We can now evaluate the line integral. We can parameterize the line segment C by x = 2 + t, y = 1 + 4t, where 0 ≤ t ≤ 1.

Substituting these into the line integral, we get:

∫CF · dr = ∫0¹ (5(2 + t), 5(1 + 4t)) · (dx, dy)

= ∫0¹ (10 + 5t, 5 + 20t) · (dx, dy)

= ∫0¹ 10 + 5t + 5 + 20t dt

= ∫0¹ 35t + 15 dt

= 145

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