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# Linear vs. exponential growth

Sal sorts various descriptions of real-world situation according to the type of growth they describe: linear or exponential. Created by Sal Khan.

Video transcript

A newborn calf weighs 40 kilograms. Each week, its weight increases by 5%. Let W be the weight in kilograms of the
calf after t weeks. Is W a linear function or an exponential
function? So, if W were a linear function, that
means that every week that goes by, the weight would
increase by the same amount. So, let's say that every week that went
by, the weight increases, or really they're talking about mass here,
the mass increased by 5 kilograms. Then we'd be dealing with a linear
function. But they're not saying that the weight
increases by 5 kilograms. They're saying by 5%. So, after one week, it'll be 1.05 times 40
kilograms. After another week, it'll be 1.05 times
that. It'll be 5% more. After the next week, it'll be 1.05 times
that. So, really what worked, if we really think
about this function, it's going to be 40 kilograms times 1.05
to the t power. We're compounding by 5% every time. We're increasing by a factor of 1.05. Another way of thinking about it by a
factor of a 105% every week. So, because we have that, that growth by a
factor, not just by a constant number, that tells us, tells us that this
is going to be an exponential function. So, let's see which of these choices
describe that. This function is linear. No, we don't have to even read that. This function is linear. Nope. This function is exponential because W
increases by a factor of 5 each time t increases by 1. No, that's not right. We're increasing by 5%. Increasing by 5% means you're 1.05 times
as big as you were before increasing. So, it's really this function is
exponential because W increases by a factor of 1.05 each time t increases
by 1. That right over there is the right answer. Let's try one more of these. Determine whether the quantity described
is changing in a linear fashion or an exponential
fashion. Fidel has a rare coin worth $550. Each year the coin's value increases by
10%. Well, this is just like the last example
we saw. We're increasing every year that goes by
as we increase by a factor of 1.1. If we grow by 10%, that's increasing by a factor of 110% or 1.1, so this is
definitely exponential. If it was increasing $10 per year, then it would be linear, but here we're increasing
by a percentage. You're uncle bought a car for 130,000
Mexican pesos. Each year, the value of the car decreases
by 10,000 pesos. So here, we're not, we're not multiplying
by a factor. We're decreasing by a fixed amount. Year one year goes by, we're at 120,000. Two years goes by, we're at 110,00. So, this is definitely a linear, this can
be described as by a linear model. The number of wild hogs in Arkansas
increases by a factor of 3 every 5 years. So, a factor of 3 every 5 years. They're not saying that it increases by 3
hogs every 5 years. We're, we're multiplying by 3 every 5
years. So, this is definitely, this one right
over here is going to be exponential. And then finally, you work as a waiter at
a restaurant. You earn $50 in tips every day you work. Well, this is super this, you know, this,
this should jump out as, as very linear. Every day you work, another $50. Work one day $50, two days $100, so forth
and so on. They're not saying you earn 50 times as
much as the day before. They're not saying that you earned 50%
more. They're saying that you're Increasing by a
fixed quantity, so this is going to be a linear model.