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The velocity, \(\mathrm{V}\) feet per second, of a model rocket \(\mathrm{t}\) seconds after launch is given by \(\mathrm{V = -32t + C}\), where \(\mathrm{C}\) is a positive constant. What is the velocity of the rocket 2 seconds after it was launched?
We need to find the velocity of the rocket at t = 2 seconds.
Given the velocity formula: \(\mathrm{V = -32t + C}\)
To find the velocity at t = 2, we’ll substitute into the formula:
\(\mathrm{V = -32(2) + C = -64 + C}\)
Since C is unknown, we need its exact value to answer the question.
Key insight: We must determine C to find the velocity at t = 2.
Statement 1: "The rocket reaches its maximum height and begins descending 1.5 seconds after it was launched."
When any object reaches its maximum height, its velocity equals zero—it momentarily stops before falling back down.
At t = 1.5 seconds:
Substituting into \(\mathrm{V = -32t + C}\):
Now we can find the velocity at t = 2:
The negative velocity indicates the rocket is descending.
[STOP - Statement 1 is Sufficient!]
Since Statement 1 allows us to find \(\mathrm{C = 48}\), we can determine the exact velocity at t = 2.
This eliminates choices B, C, and E.
Important: We now analyze Statement 2 independently, forgetting everything from Statement 1.
Statement 2: "The rocket’s initial velocity was 48 feet per second."
Initial velocity refers to the velocity at launch (t = 0).
At t = 0:
Substituting into \(\mathrm{V = -32t + C}\):
With \(\mathrm{C = 48}\), the velocity at t = 2:
[STOP - Statement 2 is Sufficient!]
Statement 2 also allows us to find \(\mathrm{C = 48}\) and determine the exact velocity at t = 2.
This eliminates choice A.
Each statement independently provides a different way to find \(\mathrm{C = 48}\):
Once we know \(\mathrm{C = 48}\), we can calculate that the velocity at t = 2 is exactly -16 feet per second.
Answer Choice D: Each statement alone is sufficient.