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# The height of a model rocket in feet, x, t seconds after its launch is

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Math Expert
Joined: 02 Sep 2009
Posts: 59568
The height of a model rocket in feet, x, t seconds after its launch is  [#permalink]

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28 Nov 2019, 00:42
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Difficulty:

55% (hard)

Question Stats:

56% (01:33) correct 44% (01:48) wrong based on 26 sessions

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The height of a model rocket in feet, x, t seconds after its launch is modeled by the equation $$x=-t^2+bt+a$$, where a and b are positive integers. At what value of t does the rocket reach its maximum height?

(1) $$b=10$$

(2) $$a=125$$

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Re: The height of a model rocket in feet, x, t seconds after its launch is  [#permalink]

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28 Nov 2019, 03:14
$$x=-t^2+bt+a$$
$$x=-t(t+b)+a$$

#1$$b=10$$
a is not know insufficient
#2
at t=0
we get height = 125
IMO B

Bunuel wrote:
The height of a model rocket in feet, x, t seconds after its launch is modeled by the equation $$x=-t^2+bt+a$$, where a and b are positive integers. At what value of t does the rocket reach its maximum height?

(1) $$b=10$$

(2) $$a=125$$

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Director
Joined: 18 May 2019
Posts: 524
Re: The height of a model rocket in feet, x, t seconds after its launch is  [#permalink]

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28 Nov 2019, 05:56
1
Given x=-t^2+bt+a
we are to determine at which time, t, x is maximum.

Given a function f(t)=mx^2+nx+k
the maximum/minimum value occurs at t=-n/ma
when m<0, then we have a maximum function and when m>0, then we have a minimum function.
From the given function x=-t^2+bt+a,
m=-1; n=b
we know that x has a maximum function since m=-1.
All we need in order to determine the time,t, at which the maximum function occurs is b. because t=-b/(2*-1) = b/2

Statement 1: b=10
This sufficient since we can determine the time that x is maximum occurs is =10/2 = 5.

Statement 2: a=125
This is insufficient since we don't b, which is necessary to determine the time, t, at which x is maximum.

Alternatively, you can use calculus to solve this question easily.
x=-t^2+bt+a
differentiating x with respect to t yields dx/dt=-2t+b
The function x is maximum when dx/dt=0
therefore -2t+b=0
t=b/2
from this we know we only need b in order to determine the time at which x is maximum.

Statement 1: b=10
so we can determine t=10/2=5
statement 1 is therefore sufficient.

Statement 2: a=125
Not sufficient, since we don't need a to determine the time at which x is maximum. We need b and we don't b.

Re: The height of a model rocket in feet, x, t seconds after its launch is   [#permalink] 28 Nov 2019, 05:56
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