# Does not compute!

• July 16th 2007, 07:03 AM
earachefl
Does not compute!
A word problem is given which I've answered correctly, but a second question doesn't add up. The problem is:

A North-South highway intersects an East-West highway at a point P. An automobile crosses P at 10:00 AM, traveling east at a constant rate of 20 mph. At that same instant another automobile is two miles north of P, traveling south at 50 mph. Find a formula which expresses the distance d between the automobiles at time t (hours) after 10:00 AM. At what time will the automobiles be 104 miles apart?

My answer, which the book confirms, is

$d = \sqrt{2900t^2 - 100t + 4}$

To answer the second question, I started with

$104 = \sqrt{2900t^2 - 100t + 4}$

$\Rightarrow 10816 = 2900t^2 -100t + 4$

$\Rightarrow 2704 = 725t^2 - 25t + 1$

$\Rightarrow 725t^2 - 25t - 2703 = 0$

Using the Quadratic Formula, I get

$\frac {25 \pm \sqrt {625 + 7838700}} {1450}$

which I can only reduce to

$\frac {5 \pm \sqrt {313573}} {290}$

But the answer given in the book is:

$\frac {1 + \sqrt 30} {29}$

So how do I get there? Did I make a mistake along the way?
• July 16th 2007, 11:47 AM
Soroban
Hello,earachefl!

Quote:

A North-South highway intersects an East-West highway at a point $P$.
An automobile crosses $P$ at 10:00 AM, traveling east at a constant rate of 20 mph.
At that same instant another automobile is two miles north of $P$, traveling south at 50 mph.
(a) Find a formula which expresses the distance $d$ between the automobiles
. . .at time $t$ (hours) after 10:00 AM.
(b) At what time will the automobiles be 104 miles apart?

I don't agree with either of their answers . . .
Code:

Q *
|
50t |
|
B *
|  *  d
2-50t |    *
|        *
- - P * - - - - - * - -
|    20t    A
|

At 10 AM, the first car is at point $P$.
. . $t$ hours later, it is at point $A\!:\;PA = 20t$

At 10 AM, the second car is at point $Q\!:\;QP = 2$.
. . $t$ hours later, it is at point $B\!:\;QB = 50t$.
. . Hence: . $BP = 2-50t$

Pythagorus says: . $d^2\:=\:(20t)^2 + (2-50t)^2$

(a) Then we have: . $\boxed{d \:=\:\sqrt{2900t^2 - {\color{red}200}t + 4}}$

If $d = 104\!:\;\;\sqrt{2900t^2 - 200t + 4} \:=\:104$

Square both sides: . $2900t^2 - 200t + 4 \:=\:10816\quad\Rightarrow\quad 2900t^2 - 200t - 10812 \:=\:0$

Divide by 4: . $725t^2 - 50t - 2703 \:=\:0$

Quadratic Formula: . $t \;=\;\frac{-(-50) \pm\sqrt{(-50)^2 - 4(725)(-2703)}}{2(725)}$

. . $t \;=\;\frac{50\pm\sqrt{7841200}}{1450} \;=\;\frac{50 \pm 20\sqrt{19603}}{1450} \;=\;\frac{5 \pm 2\sqrt{19603}}{145}$

And the positive root is: . $\boxed{t \:=\:\frac{5 + 2\sqrt{19603}}{145} \:\approx\:1.966\text{ hours.}}$