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Murljashka [212]
2 years ago
12

The position of an object along a vertical line is given by s(t) = −t3 + 3t2 + 7t + 4, where s is measured in feet and t is meas

ured in seconds. What is the maximum velocity of the object in the time interval [0, 4]?
Mathematics
2 answers:
saw5 [17]2 years ago
8 0

Answer:

The maximum velocity of the object in the time interval [0, 4] is 10 ft/s.

Step-by-step explanation:

Given : The position of an object along a vertical line is given by s(t) = -t^3+3t^2+7t +4, where s is measured in feet and t is measured in seconds.

To find : What is the maximum velocity of the object in the time interval [0, 4]?

Solution :

The velocity is rate of change of distance w.r.t time.

Distance in terms of t is given by,

s(t) = -t^3+3t^2+7t +4

Derivate w.r.t. time,

v(t)=s'(t) = -3t^2+6t+7

It is a quadratic function so its maximum is at vertex of the function.

The x point of the function is given by,

x=-\frac{b}{2a}

Where, a=-3, b=6 and c=7

t=-\frac{6}{2(-3)}

t=-\frac{6}{-6}

t=1

As 1 lie between interval [0,4]

Substitute t=1 in the function,

v(t)= -3(1)^2+6(1)+7

v(t)= -3+6+7

v(1)=10

Th maximum velocity is 10 ft/s.

Therefore, the maximum velocity of the object in the time interval [0, 4] is 10 ft/s.

FromTheMoon [43]2 years ago
5 0

Answer:

v_{max} = 10\,\frac{ft}{s} at t = 1\,s.

Step-by-step explanation:

The velocity function is found by deriving once:

v(t) = -3\cdot t^{2}+6\cdot t +7

The acceleration function is determined by deriving again:

a(t) = -6\cdot t + 6

The critical point of the velocity function is computed by equalizing the acceleration function to zero and clearing t:

-6\cdot t + 6 = 0

t = 1\,s

The Second Derivative Test is done by deriving the acceleration function and checking the critical point hereafter:

\dot a (t) = -6

Which indicates that critical point leads to maximum velocity, which is:

v(1\,s) = -3\cdot (1\,s)^{2} + 6\cdot (1\,s) + 7

v_{max} = 10\,\frac{ft}{s}

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