A plane is moving along the parabola
𝑦
=
5
−
(
2
𝑥
+
1
)
2
3
.

As the plane passes through the point
(
2
,
−
10
3
)
, its
𝑦
-coordinate is decreasing at a rate of
4.2
m/s.

At that instant, the
𝑥
-coordinate of the plane is _________________________________

Question

A plane is moving along the parabola  
𝑦
=
5
−
(
2
𝑥
+
1
)
2
3
.

As the plane passes through the point 
(
2
,
−
10
3
)
, its 
𝑦
-coordinate is decreasing at a rate of 
4.2
 m/s.

At that instant, the 
𝑥
-coordinate of the plane is _________________________________

BlackTom AI · Accepted Answer

To tackle this problem, I will treat y as a function of x along the given parabola and use the chain rule to relate dy/dt and dx/dt.

First, confirm the relation between y and x: y = 5 − ((2x + 1)^2)/3. This is a smooth function of x, so dy/dx can be found by differentiation.

Compute dy/dx: differentiate y with respect to x.
- Start with y = 5 − (1/3)(2x + 1)^2.
- Use the chain rule: d/dx of (2x + 1)^2 is 2(2x + 1)·(d/dx(2x + 1)) = 2(2x + 1)·2 = 4(2x + 1).
- Therefore dy/dx = −(1/3)·4(2x + 1) = −(4(2x + 1))/3.

Evaluate dy/dx at the given x = 2. Here 2x + 1 = 5, so dy/dx = −(4·5)/3 = −20/3 ≈ −6.6667.

Next, relate the rates with the chain rule: dy/dt = (dy/dx)·(dx/dt).
We are told the y-coordinate is decreasing at 4.2 m/s, which means dy/dt = −4.2.

Solve for dx/dt: dx/dt = (dy/dt) / (dy/dx) = (−4.2) / (−20/3) = (−4.2) · (−3/20) = 12.6/20 = 0.63.

Thus, at that instant the x-coordinate is increasing, with dx/dt = 0.63 m/s. Keeping the signs consistent with the observed directions:
- A positive dx/dt corresponds to the x-coordinate increasing.

Now, evaluate each answer option in light of this result:
- Option: decreasing at a rate of 0.63 m/s. This contradicts the computed dx/dt sign; the x-coordinate is increasing, not decreasing, so this is incorrect.
- Option: decreasing at a rate of 1.93 m/s. The magnitude 1.93 does not match the calculated 0.63, and the direction (decreasing) is wrong, so this is incorrect.
- Option: increasing at a rate of 3.82 m/s. The direction matches but the magnitude 3.82 is not what we found; this is incorrect.
- Option: increasing at a rate of 0.63 m/s. This matches both the sign (increasing) and the magnitude (0.63), so this is the correct behavior.
- Option: decreasing at a rate of 3.82 m/s. Wrong direction and wrong magnitude; hence incorrect.
- Option: increasing at a rate of 1.93 m/s. Direction is correct, but the magnitude 1.93 is not what the calculation yields, so this is incorrect.

In summary, among the given choices, the one that aligns with the chain-rule calculation is the increasing at 0.63 m/s option, while all others conflict either in sign, magnitude, or both.

A plane is moving along the parabola 𝑦 = 5 − ( 2 𝑥 + 1 ) 2 3 . As the plane passes through the point ( 2 , − 10 3 ) , its 𝑦 -coordinate is decreasing at a rate of 4.2 m/s. At that instant, the 𝑥 -coordinate of the plane is _________________________________Single choice

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