minimize word problem

[ashlee12]

What would the variables be for this word problem and what equations would I use?

A lighthouse is at point x, 4 miles offshore from the nearest point y of a straight beach. A store is at point z, 4 miles down the beach from point y. If the lighthouse keeper can row 4 miles per hour and walk 5 miles per hour how should he be proceed in order to go from the lighthouse to the store in the least possible amount of time?

 

[arthur ohlsten]

chose a point that is a units from y in the direction of z.
This is the point he will row to, and walk 4-a miles
he will row sqrt [4^2+a^2]
total time,T is

T= [4^2+a^2]^1/2 / 4 + a/5 take derivative with respect to a
dT/da= 1/8[4^2+a^2]^-1/2 [2a] +1/5
set dT/da=0
0= a/{4[16+a^2]^1/2} +1/5
0=a+.8[16+a^2]^1/2 because 4/5=.8
-a=.8[16+a^2]^1/2 square both sides
a^2=.64 [16+a^2]
a^2[1-.64] = 10.24
a^2=10.24/.36
a^2=28.4444444
a=5.3 miles impossible

he should row directly to point z and take 4sqrt2 /4 hours
1.4 hours

Arthur

 

[galactus]

Since distance=rate*time, then d/r=t.

\(\displaystyle \L\\\frac{\sqrt{x^{2}+4}}{4}+\frac{4-x}{5}\)

The derivative= \(\displaystyle \L\\\frac{x}{4\sqrt{x^{2}+16}}-\frac{1}{5}\)

Set to 0 and solve for x:

\(\displaystyle \L\\5x=4\sqrt{x^{2}+16}\)

\(\displaystyle \L\\25x^{2}=16x^{2}+256\)

\(\displaystyle \L\\9x^{2}=256\)

\(\displaystyle \L\\x^{2}=\frac{256}{9}\)

\(\displaystyle \L\\x=\frac{16}{9}\)

So, the time is minimum when \(\displaystyle \L\\x=\frac{16}{3}\approx{5.\overline{33}}\)

But, that's larger than 4, so:

\(\displaystyle \L\\\frac{\sqrt{4^{2}+4^{2}}}{4}=\sqrt{2} \;\ hrs\)