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A car, starting from rest, began moving in a straight line from a fixed point.
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Its velocity after 𝑡 seconds is given by 𝑣 equals eight 𝑡 squared plus six 𝑡 meters per second, where 𝑡 is greater than or equal to zero.
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Calculate the displacement of the car when 𝑡 equals nine seconds.
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In order to work out an expression for the displacement, we need to integrate the expression for the velocity.
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In this case, the velocity was equal to eight 𝑡 squared plus six 𝑡 this means the displacement 𝑠 will be closer to the integral of eight 𝑡 squared plus six 𝑡.
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Our limits are zero and nine as the car started from rest and we want to calculate the displacement when 𝑡 equals nine seconds.
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Integrating eight 𝑡 squared gives us eight 𝑡 cubed divided by three and integrating six 𝑡 gives us six 𝑡 squared divided by two, which can be simplified to three 𝑡 squared.
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Substituting in our limits gives us an answer of 2187.
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This means that the displacement of the car when 𝑡 equals nine seconds is 2187 meters.