Gwen's position is a function of his velocity and time
Gwen's position at 45 seconds is 63.9 feet
The function is given as:
v(t) = 250sin(t^2/120)/(t + 6)
When t = 45, we have:
v(45) = 250sin(45^2/120)/(45 + 6)
Evaluate the sum, the exponent and the quotient
v(45) = 250sin(16.875)/(51)
Evaluate the sine ratio
v(45) = 250 * 0.2903/51
Evaluate the product and quotient
v(45) = 1.42
The position is then calculated as:
S(45) = v(45) * 45
This gives
S(45) = 1.42 * 45
Evaluate the product
S(45) = 63.9
Hence, Gwen's position at 45 seconds is 63.9 feet
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The displacement is the shortest between any two given points. The displacement of Gwen at the time period of 45 seconds is 63.9 feet.
The displacement is the shortest between any two given points. It is the product of velocity and time.
Given that the function of the velocity is [tex]v(t)=\dfrac{250\sin\left(\frac{t^{2}}{120}\right)}{\left(t+6\right)}[/tex]. Therefore, substitute the value of t as 45, to know the velocity of Gwen at 45 seconds,
[tex]v(t)=\dfrac{250\sin\left(\frac{t^{2}}{120}\right)}{\left(t+6\right)}[/tex]
[tex]v(45)=\dfrac{250\sin\left(\frac{45^{2}}{120}\right)}{\left(45+6\right)}\\\\v(45) = 1.42[/tex]
Since displacement is the product of velocity and time, therefore, the displacement is equal to,
[tex]s = v \times t\\\\s = 1.42 \times 45\\\\s = 63.9\rm \ feet[/tex]
Hence, the displacement of Gwen at the time period of 45 seconds is 63.9 feet.
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