 This video is going to talk about radical applications. We know that we need to use radicals when we are solving quadratic equations, and we use them when we are solving Pythagorean theorem problems with right triangles. The radicals or functions used to model things that grow slowly over time and have a similar shape as logarithms. Suppose the graph of T of H is given where the horizontal axis, that's the x-axis, measures height and feet, and the vertical axis, that's the y, measures time and seconds for an object to fall. So if we want to estimate the starting point, it looks like it starts somewhere about approximately zero two. So what does that mean in context of the problem? Well remember we have an object that's falling, and this is the height that it falls, and this is the time it takes for the object to fall. So it sounds like, then that we are saying that we have the object falls zero feet in two seconds. So we are using the same graph, and we want to estimate T of two. Remember this is our H, which is like an x value. So that would be on the x-axis, the two, we come up here and we find this point right here. Looks like it's about the point two. It's a little more than six, so I'm going to say maybe six point one. It takes an object six point one seconds to fall two feet, according to this model. And finally, estimate H when T of H is equal to eight. So this is going to be our time. We know this is eight seconds because we want to find the height. So we look at eight on our graph, and we go across, and it looks like it's this point right here, which is right above the four. So H is equal to four, which means that it takes an object eight seconds to fall four feet. All right, let's look at another example. When an object is dropped from a height of eight feet, the time it will take to hit the ground can be approximated by this formula of T is equal to the square root of two H over 32, where T is time and seconds for the object to fall at H feet. So how long will it take an object to fall 20 feet? This is an H. 20 is an H. So we go in, we don't know what time is. It's asking us how long, so that's a T. And we have the square root of two times our H, which is 20, all over 32, which is the square root of 40 over 32. And I'm not even going to simplify that, I'm just going to go straight to my calculator and do the square root, second X square, of 40 divided by 32. Close the parentheses and we find out that it takes approximately 1.12 seconds to fall 20 feet. The last question, if an object is to fall 30 seconds, what height should the object be? Dropped from. Now the T to 30 is our T equal to the square root of two H over 32. We need to square both sides if we want to get rid of a square root. We're going to square here and we're going to square here. The square root is 900. When you square a square root, just like when we took the square root or something, a perfect square, end up with what's underneath the radical. They're inverse functions, so they undo each other. Now we just have to say 932 to clear the fraction, equal to two H, 900 times that 32, whatever that is, divided by two will be our H when we divide the two off of both sides. We can then go to our calculator and say 900 times 32 and divide it by two. We have height should be 1,400. How far is it? 1,440 feet, if it's going to take 30 seconds to drop, that's the height we need to be at.