Calculus help needed! Urgent!!!?
I really need help with this problem. I'm not getting for what I should solve and how. Can someone please help! Here is the problem:
A surveyor is standing 30 ft from the base of a building. She measures the angle of elevation to the top of the building to be 75 degrees (note degrees, not radians). How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4%.
- husoskiLv 71 decade agoBest Answer
I don't know that calculus will give you an exact answer, but it gives a pretty good answer quickly. The first trick is to write the equation in terms that you know how to take the derivative of. I'm going to use r=30 as the distance to the wall and θ~=75 as the angle. The height h is then:
h = r * tan(πθ/180)
I've explicitly converted θ from degrees to radians here because you know how to take the derivative of a tangent function that takes degrees as its input. Now, they didn't talk about errors in r, so I suppose that r is exact and h is a function of θ only. Use the chain rule to find:
h'(θ) = r * (π/180) * sec^2(πθ/180)
You can grind this out, but I'll just call it h' for now.
Suppose Δθ is the error in the measurement of θ and Δh is the corresponding error in h. If these are small, then:
Δh/Δθ ~= dh/dθ = h'
We require that Δh/h < 0.04 (for a less than 4% error) so:
0.04 > Δh/h ~= Δθ * h' / h
Δθ < 0.04 * h / h'
Notice that the units work out. h' has units of ft/deg and h is in ft. so Δθ is in degrees. You have all the info to get number answers for h and h' evaluated at r=30 ft and θ=75 deg.
- 4 years ago
Divide the integrand and then integrate: 2x/(2x+1) = 1 - 1/(2x+1) Now integrate each term to get x - 1/2 ln | 2x + 1| + c