The door chart

Dear Grandma, we have started a growth chart on the kitchen door. On every birthday someone draws a pencil line level with the top of my head and writes the date beside it. These heights are only examples.

Dear Grandma, we have a growth chart on the kitchen door: a pencil line and a date on every birthday. (Heights are examples.)

Door chart: $t$ years, $h$ cm (examples).

At five I was one hundred centimetres tall. At six, one hundred and six; at seven, one hundred and twelve; at eight, one hundred and seventeen. So I grow about six centimetres a year.

Age $t$ years, height $h$ cm: $h(5)=100,h(6)=106,h(7)=112,h(8)=117$. On average $\Delta h=\frac{117-100}{3}\approx 5.7$ a year, about $6$ cm.

$h(5)=100,h(8)=117$, so $\Delta h=\frac{117-100}{3}\approx 5.7$.

My brother said: then at thirty I will be two hundred and fifty centimetres tall! We worked it out: twenty-five more years at six centimetres is a hundred and fifty, and with the first hundred that makes two and a half metres.

My brother: at $30$, $h(30)=100+6\times 25=250$ cm!

$h(30)=100+6\times 25=250$.

That is taller than almost any door! Of course it is wrong. Growing slows down and then stops, so a straight line drawn through four marks cannot be trusted far into the future. We shall keep drawing the chart to see where the real line bends. Please come and look. With love.

$250$ cm is two and a half metres, taller than almost any door. Growth slows and stops, so a straight line from $4$ marks is wrong far ahead. We keep drawing the chart.

$250$ cm? No: growth slows, so a straight line is wrong for $t\gg 8$.