3.2 KiB
Question (Ladder questions)
A 7meter ladder leans against wall with the base
1.5meters from wall at its base. At which height does the
ladder touch the wall?
julia; hold=true; echo=false l = 7 adj = 1.5 opp = sqrt(l^2 - adj^2) numericq(opp, 1e-3)
A 7meter ladder leans against the wall. Between the
ladder and the wall is a 1m cube box. The ladder touches
the wall, the box and the ground. There are two such positions, what is
the height of the ladder of the more upright position?
You might find this code of help:
julia; eval=false @syms x y l, b = 7, 1 eq = (b+x)^2 + (b+y)^2 eq = subs(eq, x=> b*(b/y)) # x/b = b/y solve(eq ~ l^2, y)
What is the value b+y in the above?
julia; echo=false radioq(("The height of the ladder", "The height of the box plus ladder", "The distance from the base of the ladder to the box," "The distance from the base of the ladder to the base of the wall" ),1)
What is the height of the ladder
julia; hold=true; echo=false numericq(6.90162289514212, 1e-3)
A ladder of length c is to moved through a 2-dimensional
hallway of width b which has a right angled bend. If
4b=c, when will the ladder get stuck?
Consider this picture
julia; hold=true; echo=false p = plot(; axis=nothing, legend=false, aspect_ratio=:equal) x,y=1,2 b = sqrt(x*y) plot!(p, [0,0,b+x], [b+y,0,0], linestyle=:dot) plot!(p, [0,b+x],[b,b], color=:black, linestyle=:dash) plot!(p, [b,b],[0,b+y], color=:black, linestyle=:dash) plot!(p, [b+x,0], [0, b+y], color=:black)
Suppose b=5, then with b+x and
b+y being the lengths on the walls where it is stuck
and by similar triangles b/x = y/b we can solve
for x. (In the case take the largest positive value. The
answer would be the angle \theta with
\tan(\theta) = (b+y)/(b+x).
julia; hold=true; echo=false b = 5 l = 4*b @syms x y eq = (b+x)^2 + (b+y)^2 eq =subs(eq, y=> b^2/x) x₀ = N(maximum(filter(>(0), solve(eq ~ l^2, x)))) y₀ = b^2/x₀ θ₀ = Float64(atan((b+y₀)/(b+x₀))) numericq(θ₀, 1e-2)
Two ladders of length a and b criss-cross
between two walls of width x. They meet at a height of
c.
julia; hold=true; echo=false p = plot(; legend=false, axis=nothing, aspect_ratio=:equal) ya,yb,x = 2,3,1 plot!(p, [0,x],[ya,0], color=:black) plot!(p, [0,x],[0, yb], color=:black) plot!(p, [0,0], [0,yb], color=:blue, linewidth=5) plot!(p, [x,x], [0,yb], color=:blue, linewidth=5) plot!(p, [0,x], [0,0], color=:blue, linewidth=5) xc = ya/(ya+yb) c = yb*xc plot!(p, [xc,xc],[0,c]) p
Suppose c=1, b=3, and a=5.
Find x.
Introduce x = z + y, and h and
k the heights of the ladders along the left wall and the
right wall.
The z/c = x/k and y/c = x/h by similar
triangles. As z + y is x we can solve to
get
x = z + y = \frac{xc}{k} + \frac{xc}{h}
= \frac{xc}{\sqrt{b^2 - x^2}} + \frac{xc}{\sqrt{a^2 - x^2}}With a,b,c as given, this can be solved with
julia; hold=true; echo=false a,b,c = 5, 3, 1 f(x) = x*c/sqrt(b^2 - x^2) + x*c/sqrt(a^2 - x^2) - x find_zero(f, (0, b))
The answer is 2.69\dots.