Question

A spring is being pulled in a linear direction, whose motion is given by the function s(t)=kt, where k is a constant, s(t) is measured in centimeters, and t is measured in seconds for 0≤t≤65. Estimate the rate at which the spring is being pulled, with respect to time of the spring at 32 seconds in centimeters per second.

a) k
b) 32k
c) 16k
d) 0.5k

Answer 1

The rate at which the spring is being pulled is given by the derivative of the function s(t)=kt, which is simply k. Therefore, the spring's velocity at any moment, including 32 seconds, is k centimeters per second.So, the correct option is:a) k

Step-by-step explanation:

The question asks us to estimate the rate at which a spring is being pulled with respect to time, given by the function s(t)=kt, where k is a constant, and s(t) is the displacement in centimeters at time t in seconds. To find the rate, we need to compute the derivative of s(t) with respect to t, which represents the velocity of the spring or the rate of displacement per unit time.

Calculus teaches us that the derivative of s(t) with respect to t is the instantaneous rate of change of s(t), which in this case is:

ds/dt = d(kt)/dt = k

Therefore, the rate at which the spring is being pulled at any time, including at t = 32 seconds, is equal to the constant k. Hence, the correct answer to the question is (a) k centimeters per second.So, the correct option is:a) k