If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

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

Interpreting nonlinear expressions — Harder example

Video transcript

- 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.