A vehicle moves along a straight road. The vehicle's position is given by f(t), where t is measured in seconds since the vehicle starts moving. During the first 10 seconds of the motion, the vehicle's acceleration is proportional to the cube root of the time since the start. Which of the following differential equations describes this relationship, where k is a positive constant? a. df/dt= k3√t
b. df/dt= k3√f
c. d^2f/dt^2= k3√t
d. d^2f/dt^2= k3√f

The differential equation that models the situation is:

Which is option C.

The position is the integral of the acceleration, with is the integral of the acceleration, hence, the relation between the position and the acceleration is:

The acceleration is proportional to the cube root of the time since the start, hence:

In which k is the constant of proportionality.

A similar problem is given at link

C.

Step-by-step explanation:

Physically speaking, the acceleration is equal to the second derivative of the position of the vehicle in time. After a careful reading to the statement, we construct the following ordinary differential equation:

Where:

- Time, measured in seconds.

- Proportionality constant, measured in meters per second up to 7/3.

Therefore, the correct answer is C.

the given growth  function for the length of the fish (cm) is

f(m) = 5(1.07)m = 5.35m

where 

m = months

at the end of the study, the length is 9.19 cm. the  time for conducting the study is determined as

5.35m = 9.19

m = 1.7178 months

part a.

the study was performed over a time period  of about 1.72 months. therefore  the domain for  plotting the function is 0 < = m < = 1.72.

note:

if growth  continues at this rate for up to m months, then the domain would be 0< =m< =m.

part b.

the growth function is a linear function.

when m=0, f=0.

therefore the initial growth is zero, which represents the y-intercept.

part c.

assume that the growth function is valid for 0< =m< =9.

the average rate of change of the function is the slope of the linear function, which is 5.35 cm/month.