An object was thrown upward from the top of a building. The object traveled upward until it reached its maximum...

Understanding the Question

Let's understand what this question is really asking. An object is thrown upward from a building at height 38 meters, follows a parabolic path up to its maximum height, then falls to the ground. The height equation is \(\mathrm{h(t) = -4.9t^2 + bt + 38}\).

What We Need to Determine: Was the height equal to c meters at most once during the entire flight?

This is asking whether the horizontal line at height c intersects the parabola at most one point. Since we're dealing with a downward-opening parabola that starts at 38 meters and ends at ground level (0 meters), the answer depends on where c falls relative to the object's path.

Key Insight: The object starts at height 38 meters. If c is below 38, the object begins its journey already above height c—this fact will prove crucial.

For this yes/no question to be sufficient, we need to definitively answer whether \(\mathrm{h(t) = c}\) has at most one solution for positive values of t.

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{c < 38}\)

This means the target height c is below the building's top (where the object starts).

Let's visualize this step-by-step:

• The object starts at height 38 meters (already above c)
• It rises to some maximum height (moving even further above c)
• It falls back down past 38 meters and continues to the ground

Since the object starts above c and must eventually reach the ground (which is below c), when does it pass through height c?

Critical Realization: The object can only pass through height c once—during its fall. Why? Because it starts at 38 meters, which is already above c, so it never passes through c while going up.

Think of it this way: If you're on the 38th floor and throw something upward, and someone asks about the 30th floor (\(\mathrm{c < 38}\)), the object only passes the 30th floor once—on the way down.

Statement 1 is SUFFICIENT. [STOP - Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: \(\mathrm{b < c}\)

This constrains the initial velocity b to be less than c, but tells us nothing about where c is located relative to the starting height of 38.

Let's test different scenarios to see why this matters:

Scenario 1: \(\mathrm{c = 50}\) (above starting height 38)

• The object might never reach height 50 if the initial velocity isn't strong enough
• Result: \(\mathrm{h(t) = c}\) has zero solutions

Scenario 2: \(\mathrm{c = 30}\) (below starting height 38)

• The object starts above 30, rises higher, then falls through 30
• Result: \(\mathrm{h(t) = c}\) has one solution

Scenario 3: \(\mathrm{c = 42}\) (just above 38) and the object's maximum height is 45

• The object passes through 42 once while going up
• The object passes through 42 again while coming down
• Result: \(\mathrm{h(t) = c}\) has two solutions

Without knowing whether c is above, below, or between 38 and the maximum height, we cannot determine if the height equals c at most once.

Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Statement 1 alone tells us \(\mathrm{c < 38}\), which guarantees the object passes through height c exactly once (while falling). Statement 2 alone doesn't establish c's relationship to the starting height, leaving the answer uncertain.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."