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Current time:0:00Total duration:3:13

- The following equation
shows Kepler's third law of planetary motion. It relates the time, t in days,
in days, that a planet takes to revolve once around our
sun to the distance, d, in kilometers, of that
planet from the sun. So, it tells us, t how many
days, and it's in relation to the distance that that
planet, in kilometers, is from the sun. Mars is approximately four times as distant from the sun as Mercury is. About how many times longer
would Mars's revolution time be than for Mercury? Round your answer to the
nearest whole number. So, let's think about it a little bit. If you increased d by a factor of four, if you increased d by a factor of four, what's going to happen to t? Well if you increase d by a factor of four you're going to, notice you
take d to the third power. So, if you replace d with
four d, so if you replace it with four d, you're
going to get t-squared is equal to 3.98 times
ten to the negative twenty times four d to the third power. Well four d to the third power, four is going to be the same thing. I can rewrite that as four to the third, times d to the third. Four to the third power,
that's 16 times four. That's 64. So, it's gonna be the same thing as 64, times d to the third power. Or, I could just write this
as d to the third power, and then all of that times 64. Now, we have to be, we
have to be careful here. We might say hey, look, this
expression on the right side is 64 times this
expression right over here. When we increased d by a factor of four. The right side becomes 64 times as big, because remember, four to the third power, if you increase, if this becomes four d, then you have this
becomes 64 d to the third. Which we saw, right
over, right over there. But, on the left hand
side, you have a t-squared. You have a t-squared. So, over here, you would
get that t is equal to the principal root
of all of this business, right over here, the 3.98 times ten to the negative twenty
times d to the third. Here you're gonna get
t is going to be equal to the principal root of, t is going to be the principal root of 3.98 times ten to the negative twenty
times d to the third, times, you could throw
the 64 under the radical, or you could say times
the principal root of 64. Well the principal root of 64 is eight. So, times eight. So, when you look at it this way, you increase d by factor of four, you're going to increase
t by a factor of eight. This expression is the same
thing as this expression, but we have a times, we
have a times eight here now. So, about how many times longer would Mars's revolution time
be than for Mercury? Well, eight times longer.