A small submarine is in a narrow underwater canyon 30 feet above the bottom searching for two sunken ships that are known to be 500 feet apart on the canyon floor. The sonar indicates that one of the wrecks is on the bottom in front of the sub, 475 feet from the nose of the sub, and the other wreck is on the bottom directly behind the sub, 75 feet from the nose. The difference in the distance from the wrecks to the sub is constant on a hyperbola having the wrecks as foci. Assume the two wrecks are positioned on a rectangular coordinate system at points with coordinates (-250, 0) and (250, 0) as shown in the figure. Find the x-coordinate of the position of the submarine. (Round to the nearest whole number, if necessary.)
A) x = 150
B) x = 250
C) x = 38,400
D) x = 200
E) x = 204

Question

A small submarine is in a narrow underwater canyon 30 feet above the bottom searching for two sunken ships that are known to be 500 feet apart on the canyon floor. The sonar indicates that one of the wrecks is on the bottom in front of the sub, 475 feet from the nose of the sub, and the other wreck is on the bottom directly behind the sub, 75 feet from the nose. The difference in the distance from the wrecks to the sub is constant on a hyperbola having the wrecks as foci. Assume the two wrecks are positioned on a rectangular coordinate system at points with coordinates (-250, 0) and (250, 0) as shown in the figure. Find the x-coordinate of the position of the submarine. (Round to the nearest whole number, if necessary.) 
A) x = 150 
B) x = 250 
C) x = 38,400 
D) x = 200 
E) x = 204

Quizplus · Accepted Answer

We can draw a diagram to represent the problem:

[asy]
unitsize(0.02 cm);

draw((0,-10)--(0,10), Arrow);
draw((-15,0)--(15,0), Arrow);

label("$x$", (15,0), E);
label("$y$", (0,10), N);

draw((-13,-8)--(13,8), red);
dot((-8,0));
dot((8,0));
draw((-13,-10)--(-8,0)--(-13,10), dashed);
draw((13,-10)--(8,0)--(13,10), dashed);

dot((8,-3));
dot((-8,3));

label("$(250, 0)$", (250,0), E);
label("$(-250, 0)$", (-250,0), W);
label("$S$", (8,-3), NE);
label("$W_2$", (13,-9), SE);
label("$W_1$", (-13,9), NW);

draw((-250,0)--(8,-3));
draw((250,0)--(-8,3));
draw((8,-3)--(8,0),dashed);
draw((-8,3)--(-8,0),dashed);
draw((0,-3)--(8,-3),dashed);
draw((0,3)--(-8,3),dashed);

label("475", (180,-2));
label("75", (-90,2));
label("$a$", (-4,-1.5));
[/asy]

We are looking for the x-coordinate of the submarine, which we will call $S$. We can use the information given to set up equations involving the distances from $S$ to each wreck. Let $W_1$ and $W_2$ be the positions of the wrecks as shown in the diagram. Then:

$$SW_1 - SW_2 = k$$
where $k$ is a constant. We know that $SW_1 = \sqrt{(x + 250)^2 + 9}$ and $SW_2 = \sqrt{(x - 250)^2 + 9}$ (using the Pythagorean theorem), so we can substitute:

$$\sqrt{(x + 250)^2 + 9} - \sqrt{(x - 250)^2 + 9} = k$$

We are given two more pieces of information: $SW_1 = 475$ and $SW_2 = 75$. Using these and simplifying the previous equation, we get:

$$(x + 250)^2 - (x - 250)^2 = (475 + 75 + k)(475 - 75 - k)$$

Simplifying further, we get:

$$500x = 4k(400 + k)$$

We can see that $k$ must be positive (since $SW_1 > SW_2$), so $4k(400 + k)$ is an increasing function. Therefore, the only possible value of $x$ is when $4k(400 + k)$ is closest to 500$x$.

When $k = 75$ (corresponding to $SW_2$ being smallest), we get $4k(400 + k) = 150000$. When $k = 200$, we get $4k(400 + k) = 240000$. Therefore, we want $k$ to be closer to 200. We can estimate the answer by using $k = 175$ (which gives $4k(400 + k) = 218500$) and $k = 225$ (which gives $4k(400 + k) = 261000$). Since the actual value of $k$ is closer to 200, we want a value of $x$ that gives $k$ closer to 200.

If $x = 203$, then $SW_1 = \sqrt{(203 + 250)^2 + 9} \approx 475.11$ and $SW_2 = \sqrt{(203 - 250)^2 + 9} \approx 74.89$, which gives $k = 400 - 75 \approx 325$. Using this value of $k$, we get $4k(400 + k) \approx 260800$, which is closest to 203 * 500 = 101500. Therefore, the final answer is $\boxed{	ext{(E) }x = 204}$.

A Small Submarine Is in a Narrow Underwater Canyon 30