Final answer:
To determine if the pilot can hit the target at the origin, we need to find the values of t that satisfy the equation r(t) = <5-t, 21-t², 3- t³/27> = <0, 0, 0>. The pilot can hit the target at exactly one time, t = 5. The distance between the pilot and the target at this time is approximately 11.44 units.
Step-by-step explanation:
In order to determine if the pilot can hit the target at the origin, we need to find the values of t that satisfy the equation r(t) = <5-t, 21-t², 3- t³/27> = <0, 0, 0>.
To do this, we can set each component of r(t) equal to zero and solve for t.
The equation for the x-component gives us t = 5, the equation for the y-component gives us t = ± √21, and the equation for the z-component gives us t = -3. Since the question specifies that the pilot can hit the target only once, we choose the solution t = 5.
To find the distance between the pilot and the target at this time, we can evaluate the magnitude of r(t) at t = 5.
The distance is given by the formula |r(t)| = sqrt((5-0)² + (21-25)² + (3-125/27)²) = sqrt(1152/27) ≈ 11.44 units.
Using forward lasers rather than rear lasers makes sense in this scenario because the fighter plane is traveling along the path defined by r(t), which means the target at the origin is ahead of the plane in the forward direction. Therefore, the pilot needs to use forward lasers to hit the target.