Download Math 1300 3.8 Word Problem Name: Solutions On roads on steep

Math 1300
3.8 Word Problem
Name: Solutions
On roads on steep hills, you sometimes see a sign that reads “Steep hill: 20% Grade”. The grade
of a hill is the slope (rise/run) written as a percentage, or, equivalently, as the number of feet the
hill rises vertically per hundred feet horizontally.
(a) Let x be the grade of a hill. Explain why the angle, θ degrees, that a hill makes with the
horizontal is given by
x 180
θ=
arctan
π
100
Since tangent equals opposite over adjacent, we can see from thepicture
(which was on the
x
x handout) that tan(θ) =
if θ is in radians, giving θ = arctan
. Since we need θ in
100
100
180
to get the formula proposed above.
degrees, we need to multiply by the conversion factor
π
dθ
dθ
in terms of x. Then find
for grades of x = 0%, 10%, and 20%.
dx
dx
We use the chain rule:
(b) Find an equation for
dθ
180
1
1
9
1
=
·
=
·
·
dx
π 1 + x 2 100
5π 1 + x 2
100
100
Now we plug in, using a calculator, to get
dθ
dθ
dθ
|x=0 = 0.5729578
|x=10 = 0.56728495
|x=20 = 0.55092096
dx
dx
dx
dθ
(c) You can estimate θ at x = 20% simply by multiplying
at x = 0 by 20. How could you
dx
write this estimate as a linear approximation? How much error is there in the value of θ found
by using this method rather than by using the exact formula that involves the inverse tangent
function?
The tangent line approximation to θ near x = 0 is
θ(x) ≈ 0.5729578x
The error is
E(x) =
x 180
arctan
− 0.5729578x
π
100
The error when x = 20 is
E(20) =
180
arctan
π
20
100
− 0.5729578(20) = −0.1492234 degrees
The error when x = 100 is
180
E(100) =
arctan
π
100
100
− 0.5729578(100) = −12.29578 degrees
Math 1300
3.8 Word Problem
Name: Solutions
(d) A rule of thumb you can use to estimate the number of degrees a hill makes with the horizontal
is to divide the grade by 2. Where in your work for part (c) did you divide by approximately
2? When you use this method to determine the number of degrees for grades of 20% and 100%,
how much error is there in the number?
We divided by approximately 2 when we multiplied x by 0.5729578.
Using this rule of thumb at x = 10, θ(20) ≈ 20/2 = 10. This gives an error of
180
20
− 10 = 1.3099 degrees
arctan
π
100
.
When x = 100, this “rule of thumb” gives an error of exactly −5 degrees.