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Scilla [17]
3 years ago
7

Given s left parenthesis t right parenthesis equals 5 t squared plus 5 ts(t)=5t2+5t​, find

Mathematics
1 answer:
finlep [7]3 years ago
6 0

Answer:

The velocity function is v(t)=10t+5.

The acceleration function is a(t)=10.

When t = 44​, the velocity is v(44)=445 \:\frac{ft}{s}.

When t = 44​, the acceleration is a(44)=10\: \frac{ft}{s^2}.

Step-by-step explanation:

We know that the position function is given by

s(t)=5t^2+5t

Velocity is defined as the rate of change of position or the rate of displacement. If you take the derivative of the position function you get the instantaneous velocity function.

v(t)=\frac{ds}{dt}

Acceleration is defined as the rate of change of velocity. If you take the derivative of the instantaneous velocity function you get the instantaneous acceleration function.

a(t)=\frac{dv}{dt}

The instantaneous velocity function is given by

v(t)=\frac{d}{dt} s(t)=\frac{d}{dt}(5t^2+5t)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\v(t)=\frac{d}{dt}\left(5t^2\right)+\frac{d}{dt}\left(5t\right)\\\\\mathrm{Apply\:the\:Power\:Rule}:\quad \frac{d}{dx}\left(x^a\right)=a\cdot x^{a-1}\\\\v(t)=10t+\frac{d}{dt}\left(5t\right)\\\\\mathrm{Apply\:the\:common\:derivative}:\quad \frac{d}{dt}\left(t\right)=1\\\\v(t)=10t+5

The instantaneous acceleration function is given by

a(t)=\frac{dv}{dt} =\frac{d}{dt}(10t+5)\\\\\mathrm{Apply\:the\:Sum/Difference\:Rule}:\quad \left(f\pm g\right)'=f\:'\pm g'\\\\a(t)=\frac{d}{dt}\left(10t\right)+\frac{d}{dt}\left(5\right)\\\\a(t)=10

To find the velocity and acceleration when t = 44, we substitute this value into the velocity and acceleration functions

v(44)=10(44)+5\\v(44)=445 \:\frac{ft}{s}

a(44)=10\: \frac{ft}{s^2}

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