### Instruction: A particle moves so that it is at $(2t + 7, 4t - 13)$ at time $t.$  Find the speed of the particle, measured in unit of distance per unit of time.### Response: The speed of the particle is the magnitude of its velocity vector, which is the derivative of its position vector.
To find the velocity vector, I need to differentiate the components of the position vector with respect to $t.$
The derivative of $2t + 7$ is $2,$ and the derivative of $4t - 13$ is $4,$ so the velocity vector is $(2, 4).$
The magnitude of the velocity vector is the square root of the sum of the squares of its components, by the Pythagorean theorem.
The square of $2$ is $4,$ and the square of $4$ is $16,$ so the sum of the squares is $20.$