Can you check what I did and help me with the third question? Thanks for the help!
v(t)=25tsin(t^2)
At time=0, you were backed up against the bumper of your car.
How far were you from your car two minutes after you started walking?
Integral (0 to 2) v(t) dt = 20.671
What was your acceleration 1 minute into your walk?
V’(1)=48.052
At what time was your instantaneous velocity as your average velocity over the first minute? (how do I solve this?)
Your V'(1) looks OK.
It is 50 cos 1 + 25 sin 1
I didn't check your integral.
Your wording of the last question, with the word "as", doesn't make sense to me. Do you mean to ask
"At what time was your instantaneous velocity equal to your average velocity over the first minute?"
To answer this, set
v(t) = (distance traveled)/t = x(t)/t
Use your x(t) solution to get that.
To find the time when your instantaneous velocity is equal to your average velocity over the first minute, you need to set up an equation and solve for the time.
Here's how you can do it step by step:
Step 1: Determine the average velocity over the first minute.
To calculate the average velocity, you need to find the displacement over the first minute and divide it by the time elapsed. In this case, the displacement is given by the integral of the velocity function from 0 to 1, and the time elapsed is 1 minute.
Average Velocity = (1 / 1) * Integral (0 to 1) v(t) dt
Step 2: Set up the equation.
Let's denote the time when the instantaneous velocity is equal to the average velocity as t' (unknown). Now, we need to find when the instantaneous velocity is equal to the average velocity.
v(t') = Average Velocity
Step 3: Solve the equation.
To solve the equation, substitute the expression for the velocity function (v(t) = 25tsin(t^2)) into the equation and solve for t'.
25t'sin((t')^2) = Average Velocity
Once you have the equation, you can use algebraic or numerical methods to solve for t'.
Keep in mind that since the equation involves trigonometric functions and the solution may not have an exact closed form, you might need to use numerical methods like approximation or graphing to find the value of t'.
I hope this helps you solve the third question. If you have any more questions or need further clarification, feel free to ask!