Sharpening the axe

I was cutting logs for the winter, eight hours a day. With my blunt axe I cut ten logs an hour.

A day of $8$ h, a blunt axe: $10$ logs an hour.

Day $T=8$ h, blunt $r_b=10$.

A friend said: "Stop and sharpen the axe." I said I had no time. "It takes an hour," she said, "and then you will cut fifteen logs an hour."

Sharpening takes $1$ h, and then $15$ logs an hour.

Sharpening $s=1$ h, sharp $r_s=15$.

I did the sum. Without sharpening: eight hours, ten logs each, eighty logs. With sharpening: seven hours of cutting, fifteen logs each, a hundred and five logs. So I sharpened, and cut twenty-five logs more.

Blunt: $8\times 10=80$. Sharp: $(8-1)\times 15=105$. The difference is $105-80=25$ more logs.

$8\times 10=80<105=7\times 15$

From which day length does sharpening pay? When the day is longer than three hours. At exactly three hours the two ways tie: thirty logs each. On a short day, do not bother; on a long day, stop and sharpen.

For a day of $T$ hours, blunt gives $10T$ and sharp gives $15(T-1)$. Sharpening pays when $15(T-1)>10T⟺T>3$, and for $T=3$ both give $30=30$.

$15(T-1)>10T⟺T>3$; $T=3$ gives $30=30$