Posted by Aletha on Sunday, March 18, 2007 at 6:51pm.
After t hours of an 8-hour trip the distance a car travels is modeled by:
D(t)= 10t + (5)/(1+t) - 5
where D(t) is measured in meters.
a) derive a formula for the velocity of the car.
b) how fast is the car moving at 6 hours?
c) derive a formula for the car's acceleration.
My answers:
a)(5)/(1+t)^2
b).102 meters (I KNOW this is wrong!)
c) ????
I just know my answers are wrong if someone could just tell me where I went wrong and how do you solve c).
v=d D/dt= 10 -5/(1+t)^2
acceleration is the first derivative of v, use the v above not yours.
Is b) correct?
To derive a formula for the velocity of the car (v), you need to take the derivative of the distance equation (D(t)) with respect to time (t).
a) The derivative of D(t) is given by:
v(t) = dD(t)/dt = 10 - 5/(1+t)^2
So, the formula for the velocity of the car is v(t) = 10 - 5/(1+t)^2.
b) To find how fast the car is moving at 6 hours (t=6), you need to substitute the value of t into the velocity formula:
v(6) = 10 - 5/(1+6)^2
v(6) = 10 - 5/49
v(6) = 9.898 meters per hour (accurate to three decimal places)
c) To derive a formula for the car's acceleration (a), you need to take the derivative of the velocity equation (v(t)) with respect to time (t).
The derivative of v(t) is given by:
a(t) = dv(t)/dt = d²D(t)/dt² = 10 + 10(5)/(1+t)^3 = 10 + 50/(1+t)^3
So, the formula for the car's acceleration is a(t) = 10 + 50/(1+t)^3.
Regarding your answer for b), it seems to be incorrect. The correct answer is approximately 9.898 meters per hour.