A guided missile is fired from an airplane. The displacement $d$ of the missile from the surface of the sea in kilometers at time $t$ in seconds is given by the equation $d = \frac{6}{t-2} - 1$. The missile is fired 3 seconds after the order is given, and at that time, the missile is 5 km above the sea. We are asked to: (1) Find how long it will take for the missile to be 1 km above the surface of the sea. (2) Find after how many seconds the missile will hit the surface of the sea. (3) Explain why the missile will never be displaced more than 1 km below the surface of the sea.

1. Problem Description

A guided missile is fired from an airplane. The displacement $d$ of the missile from the surface of the sea in kilometers at time $t$ in seconds is given by the equation $d = \frac{6}{t-2} - 1$. The missile is fired 3 seconds after the order is given, and at that time, the missile is 5 km above the sea. We are asked to:

(1) Find how long it will take for the missile to be 1 km above the surface of the sea.

(2) Find after how many seconds the missile will hit the surface of the sea.

(3) Explain why the missile will never be displaced more than 1 km below the surface of the sea.

2. Solution Steps

(1) To find the time when the missile is 1 km above the sea, we set $d=1$ and solve for $t$.

$1 = \frac{6}{t-2} - 1$

$2 = \frac{6}{t-2}$

$2(t-2) = 6$

$2t - 4 = 6$

$2t = 10$

$t = 5$

The time is $t=5$ seconds. Since the missile is fired 3 seconds after the order is given, this is $5$ seconds from $t=0$.

(2) To find the time when the missile hits the surface of the sea, we set $d=0$ and solve for $t$.

$0 = \frac{6}{t-2} - 1$

$1 = \frac{6}{t-2}$

$t-2 = 6$

$t = 8$

The time is $t=8$ seconds. Since the missile is fired 3 seconds after the order is given, this is $8$ seconds from $t=0$.

(3) We need to analyze the equation $d = \frac{6}{t-2} - 1$.

For the missile to be more than 1 km below the surface of the sea, we would need $d < -1$.

$\frac{6}{t-2} - 1 < -1$

$\frac{6}{t-2} < 0$

Since 6 is positive, we need $t-2 < 0$, so $t < 2$.

However, the missile is fired 3 seconds after the order is given, so we must have $t \ge 3$.

Thus $t<2$ and $t\ge 3$ cannot occur simultaneously.

Alternatively, consider the limit of $d$ as $t$ approaches infinity.

$\lim_{t\to\infty} \frac{6}{t-2} - 1 = 0 - 1 = -1$.

So, as time goes to infinity, the displacement $d$ approaches $-1$. Thus, the displacement $d$ will never be less than $-1$.

3. Final Answer

(1) It will take 5 seconds for the missile to be 1 km above the sea.

(2) The missile will hit the surface of the sea after 8 seconds.

(3) The missile will never be displaced more than 1 km below the surface of the sea because $d$ approaches -1 as $t$ goes to infinity, and $t$ must be greater than or equal to

3. Furthermore, for $d$ to be less than -1, $t$ must be less than 2, but the missile is fired at $t=3$, meaning $t$ cannot be less than

2.