The differential equation below models the temperature of an 89C cup of coffee in a 23C room, where it is known that the coffee cools at a rate of 1C per minute when its temperature is 73C. Solve the differential equation to find an expression for the temperature of the coffee at time t. (Let y be the temperature of the cup of coffee in C, and let t be the time in minutes, with

t = 0 corresponding to the time when the temperature was 89C.)

formula:

dy

dt

=

1

50

(y 23)

2. Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the parameter.

x = 2 + ln(t), y = t² + 1, (2, 2)

y=?

3. At what point on the curve x = 9t² + 3, y = t³ 7 does the tangent line have slope 1/2?

(x,y)=?

4. Find the exact length of the curve.

x = 2 + 12t², y = 1 + 8t³, 0 t 2