yrozenblum wrote:
An object was thrown upward from the top of a building. The object traveled upward until it reached its maximum height and then fell until it hit the ground next to the building. Between the time the object was thrown and when the object hit the ground, its height above level ground was modeled by the equation \(h(t) = -4.9t^2 + bt + 38\), where \(h(t)\) is the height, in meters, \(t\) is the number of seconds after the object was thrown, and \(b\) is a positive constant. During this time, was the height of the object above the ground equal to \(c\) meters at most once?
(1) \(c < 38\)
(2) \(b < c\)
At t = 0, the height of the object = the height of the building
\(h(0) = -4.9t^2 + bt + 38 = 38\)
The object traveled upward until it reached its maximum height and then fell until it hit the ground next to the buildingHence, we can infer that the height of the object increased and then decreased, and finally, when it hit the ground, the height of the object was zero.
The object will attain a height more than once if that height is part of its onward and return journey. Put other words, if \(c\) is between 38 and the maximum height of the object , the object will attain the height '\(c\)' meters above ground twice. In all other cases, the object will attain the height 'c' meters above ground at most once.
Statement 1(1) \(c < 38\)
The height of the object first increased and then decreased. As \(c\) is between \(0\) and \(38\), we can infer that at some point in time when the object was traveling downwards, the object must have attained a height equal to \(c\) meters. In its journey, the object attained this height only once.
The response to the question "Was the height of the object above the ground equal to \(c\) meters at most once?" is Yes.
The statement is sufficient. We can eliminate B, C, and E.
Statement 2(2) \(b < c\)
We don't know the maximum height reached by the object. Assume that the maximum height reached by the object is \(40\) meters and the value of \(c\) is 39 meters, we can conclude that the object reached the value of \(c\) meters above the ground twice during its journey, once when the object moved upwards and once when the object moved downwards.
In that case, the response to the question "Was the height of the object above the ground equal to \(c\) meters at most once?" is No.
However, if the maximum height reached by the object is \(40\) meters and the value of \(c\) equals \(31\) meters, we can infer that the object attainted that height only once during the downward journey.
In that case, the response to the question "Was the height of the object above the ground equal to \(c\) meters at most once?" is Yes.
Hence, statement 2 is not sufficient. We can eliminate D.
Option A