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28 Nov 2019, 00:42
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A
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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) \(a=125\)
Are You Up For the Challenge: 700 Level Questions
(1) \(b=10\)
(2) \(a=125\)
Are You Up For the Challenge: 700 Level Questions
28 Nov 2019, 03:14
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\(x=-t^2+bt+a\)
\(x=-t(t+b)+a\)
#1\(b=10\)
a is not know insufficient
#2
at t=0
we get height = 125
IMO B
\(x=-t(t+b)+a\)
#1\(b=10\)
a is not know insufficient
#2
at t=0
we get height = 125
IMO B
Bunuel
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/2m
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 is sufficient since we can determine the time t that x is maximum, t=10/2 = 5.
Statement 2: a=125
This is insufficient since we don't know b, which is necessary to determine the time, t, at which x is maximum.
The answer therefore is option A.
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 only need b and we don't have b.
The answer is therefore 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/2m
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 is sufficient since we can determine the time t that x is maximum, t=10/2 = 5.
Statement 2: a=125
This is insufficient since we don't know b, which is necessary to determine the time, t, at which x is maximum.
The answer therefore is option A.
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 only need b and we don't have b.
The answer is therefore A.
