2. ## Re: Shortest possible diatnace to link these four towns

The lowest I've been able to get is 652.93 miles and a final answer of 19588 million £

3. ## Re: Shortest possible diatnace to link these four towns

I get a substantially higher minimum figure. I don't see how you can possibly get an answer that is not a integer number of miles - please explain.

4. ## Re: Shortest possible diatnace to link these four towns

I linked the towns in this kind of fashion

A.B
\ ____________ /
/ \
BC
Ignore the

Then I found the minimum if this function :

4(√(106.5^2+n^2))+(284-2n), 0≤n≤142

5. ## Re: Shortest possible diatnace to link these four towns

I get 21600 million dollars.

355^2 = 213^2 + 284^2
ABCD (labelled by beginning letters of each town) is a rectangle.

The shortest road would be a cross through the rectangle.. so 355 + 355 = 710 miles
OR

AD (213 miles) + DC (284 miles) + CB (213 miles)
or a similar variation
DA (213 miles) + AB (284 miles) + BC (213 miles)

In any case.. the combined length of roads (minimum) seems to be 710 miles.

I don’t see how it could be less than that.

6. ## Re: Shortest possible diatnace to link these four towns

road miles, $m$ ...

$m = 4\sqrt{x^2+106.5^2} + (284-2x)$

$\dfrac{dm}{dx} = \dfrac{4x}{\sqrt{x^2+106.5^2}} - 2$

$\dfrac{dm}{dx} = 0 \implies \dfrac{2x}{\sqrt{x^2+106.5^2}} = 1$

$4x^2 = x^2+106.5^2$

$x = \dfrac{106.5}{\sqrt{3}} \implies m \approx 653$ miles

7. ## Re: Shortest possible diatnace to link these four towns

thanks for your guide...... its the best explain