linear approximation - triangle

Question

The length L of a long wall is to be approximated. The angle θ, as shown in the diagram (not to scale), is measured to be 45.5∘, accurate to within 0.2∘

∘. Assume that the triangle formed is a right triangle.

image: https://webwork.uoregon.edu/webwork2_files/tmp/Math251-25349/images/f36cfca6-19eb-3877-bb32-b0394831da9d___d056c5ad-5d1d-33ac-8d58-c51beb770d78.jpg

(a) What is the measured length of the wall? ___ ft

(b) Estimate an upper bound for the propagated error using a linear approximation? ___ ft

(c) What is the upper bound for the percent error? ___%

please format answers in (a-c) format. thanks!

Answer 1

Hi Kate,

First you want to compute the length L based on the measured angle:

L = 25 tan(θ)

= 25 tan(45.5°)

≈ 25.4 ft

Next, you need to look at the error propagation. It is very important you know that whenever doing calculus with trig functions, you must use radians. From the above equation, taking the differential, and use the fact that, after converting .2° to radians we have dθ = .00349 rad, and we convert the angle as well 45.5° = .7941 rad, we get

dL = 25 sec²(θ) dθ

≈ 25 sec²(.7941) .00349

≈ .1776

so the upper bound for propagated error would be .1776 ft. Then the upper bound for percent error would be

percent error = (dL / L) · 100

= (.1776 / 25.4) · 100

≈ .6992

I am not sure what is meant by (a-c) format, perhaps that is something specific to your class, but in any case, I hope that is helpful!