How much work is needed to extend a 20 foot fence so that one end touches the ground. The fence as a linear mass density of (.4x - .02x^2) lb./ft.
To find the amount of work needed to extend the fence, we can use the formula for work:
Work = ∫[a, b] F(x) dx
Where F(x) represents the force at each point along the fence, and [a, b] represents the interval over which we are calculating the work.
In this case, the force at each point along the fence is the product of the linear mass density (λ) and the gravitational acceleration (g). Therefore, F(x) = λ(x) * g.
Given that the linear mass density is (0.4x - 0.02x^2) lb./ft, and assuming a constant gravitational acceleration of 32.2 ft./s^2, we can substitute these values into the equation for F(x):
F(x) = (0.4x - 0.02x^2) * 32.2 lb./ft.
Next, we need to determine the interval [a, b]. In this case, we are extending the fence so that one end touches the ground. Let's assume the initial length of the fence is L. Therefore, b = L, and a = L - 20, as we want to extend the fence 20 additional feet.
Finally, we can calculate the work by integrating F(x) over the interval [a, b]:
Work = ∫[L-20, L] (0.4x - 0.02x^2) * 32.2 dx lb.
To solve this integral, you can use calculus techniques or a symbolic math software. After evaluating the integral, you will obtain the amount of work needed to extend the fence in lb.