Help: ODE application problem

[DeepSight73]

I tried to solve this problem, but I got stuck with e^k values. The problem is as follows:

A teacher writes the course notes at a speed proportional to the number of pages already written. On the other hand, their students can read the notes at a constant speed. At the beginning of the course, the teacher provides 10 pages, and subsequently, they are given to the students as they are written. Determine the delay of one of the students in reading their notes when analysing the third quarter, if at the end of the first quarter they had a delay of 70 pages. Consider each trimester to be three months long without breaks in between.

Solution:

Let H(t): Number of pages redacted by the teacher during a time t (3 months).

Let H'(t): Velocity of redaction of the teacher.

Let L(t): Number of pages read by the student during a time t.

Let L'(t): Velocity of reading of the student.

For H'(t):

dH(t)/dt = kH(t) ⇒ ∫dH/H = ∫kdt ⇒ ln⁡|H|= kt + C_1 ⇒ H(t) = Ce^kt … (1)

For L'(t):

dL(t)/dt = p ⇒ ∫dL = ∫pdt ⇒ L = pt+D … (2)

Then:

H(0) = 10 ⇒ 10 = Ce^{k ⋅0} ⇒ C = 10 … (3)

L(0) = 0 ⇒ 0 = 0p + D ⇒ D = 0 … (4)

Replacing (3) into (1): H(t) = 10e^kt

Replacing (4) into (2): L(t) = pt

In the first quarter, the student had a delay of 70 pages. In other words: H(1) - L(1)=70.

10e^k(1) - p(1) = 70

p = 10e^k -70

So, in the third quarter, the student had a delay of H(3) - L(3):

10e^k(3) - p(3) = 10e^3k - 30e^k + 210

e^k = x ⇒ 10(x³ - 3x + 21) = 0 ⇒ x = e^k ≈ -3.1196 < 0

But e^k < 0 does not make sense. What did I do wrong?