Integrals and exponential growth problem

[ringo123]

Homework Statement

Before the AP exam Cal Q Luss has 3 hours to cram: during this time, he wants to memorize a set of 60 derivative/integral formulas. According to psychologists, the rate at which a person can memorize a set of facts is proportional to the number of facts remaining to be memorized . Thus if he memorizes y facts in t minutes, the model would be:
dy/dt = k(60-y) where k is a positive constant

Initially, Cal knows no facts.

A. Write an equation for y as a function of t.
B. If he memorizes 15 formulas in the first twenty minutes, how many facts will he memorize in:
a. 1 hour
b. 3 hours

Homework Equations

I'm guessing A=Pe^rt?

The Attempt at a Solution

I did separation of variables and I got that

-y-60=e^kt

but I don't know what to do after that.

 

[Mute]

Homework Statement

Before the AP exam Cal Q Luss has 3 hours to cram: during this time, he wants to memorize a set of 60 derivative/integral formulas. According to psychologists, the rate at which a person can memorize a set of facts is proportional to the number of facts remaining to be memorized . Thus if he memorizes y facts in t minutes, the model would be:
dy/dt = k(60-y) where k is a positive constant

Initially, Cal knows no facts.

A. Write an equation for y as a function of t.
B. If he memorizes 15 formulas in the first twenty minutes, how many facts will he memorize in:
a. 1 hour
b. 3 hours

Homework Equations

I'm guessing A=Pe^rt?

The Attempt at a Solution

I did separation of variables and I got that

-y-60=e^kt

but I don't know what to do after that.

You're close, but your solution is not quite right. For one, you made a minus sign mistake: it should still be -(60-y) or -(y-60), depending on which of y or 60 is initially greater, not -y-60. This error will affect the sign of your exponentially, too. You also forgot the constant of integration.

Once you've fixed those errors, you need to use the data you are given to figure out the values of the constants you don't know.

 

[Ray Vickson]

Homework Statement

Before the AP exam Cal Q Luss has 3 hours to cram: during this time, he wants to memorize a set of 60 derivative/integral formulas. According to psychologists, the rate at which a person can memorize a set of facts is proportional to the number of facts remaining to be memorized . Thus if he memorizes y facts in t minutes, the model would be:
dy/dt = k(60-y) where k is a positive constant

Initially, Cal knows no facts.

A. Write an equation for y as a function of t.
B. If he memorizes 15 formulas in the first twenty minutes, how many facts will he memorize in:
a. 1 hour
b. 3 hours

Homework Equations

I'm guessing A=Pe^rt?

The Attempt at a Solution

I did separation of variables and I got that

-y-60=e^kt

but I don't know what to do after that.

Before trying to do anything, stop and perform a simple reality test. If you put t = 0 in your solution you get y = -60; this cannot be right, because y ≥ 0. So, you made an error somewhere, and trying to go further with your present "solution" would only make things worse.

You need to go back and check your work.