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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

Voiceover: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%. 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 percent more. After the next week
it'll be 1.05 times that. So really, 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. Or another way of thinking
about it, by a factor of 105% every week. Because we have that growth by a factor, not just by a constant
number, that tells us that this is going to be
an exponential function. So let's see which if 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 1 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. Your 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 multiplying by a factor, we're decreasing by a fixed amount. 1 year goes by, we're at 120,000. 2 years goes by we're at 110,000. So this is definitely a linear ... This can be described 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 it increases by 3 hogs every 5 years. We're multiplying by 3 every 5 years. So this is definitely ... This one right over here
is going to be exponential. Then, finally, you work as
a waiter at a restaurant. You earn $50 in tips every day you work. Well, this is super ... This should jump out as very linear. Every day you work, another $50. Work 1 day, $50. 2 days, $100. So forth and so on. They're not saying that
you earned 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.