{Use of Tech} Height and time The height in feet of a baseball...

Question

{Use of Tech} Height and time The height in feet of a baseball hit straight up from the ground with an initial velocity of 64 ft/s is given by h= ƒ(t) = 64t - 16t² where t is measured in seconds after the hit.

e. At what time is the ball at a height of 10 ft on the way down?

e. At what time is the ball at a height of 10 ft on the way down?

Accepted Answer

Welcome back everyone. In this problem, a share market opens at 9 AM and closes at 3 PM. The price of a share for a particle share on a given date is defined by the function p of x equals 12 x minus 2 x squared, where x is the number of hours from 9 AM. And we want to find the value of x when the value of the price of a particle share, p of x, is $10 and p of x is decreasing. For our answer choices, a is 1, b is 2, c is 3, and d is 4. Now, let's let's think about our problem a little bit. What are we trying to figure out? Well, we're given a function p of x and that tells us the price of a share for a particle share given the number of hours. So in other words, our function takes the number of hours x, applies the function p, and then we get an output p of x. But, in our problem, we are given the value of p of x. We're given the price and now we want to figure out the value of x. In other words, the number of hours from 9 AM. So, it's as if we're doing the inverse of our function. Now, given the price, we want to find the number of hours. So since we're undoing our function, we're doing the opposite. It means if we can rewrite p of x to find its inverse, then we can go ahead and solve for the value of x. So let's go ahead and do that. So, we know that our function is p of x equals 12 x minus 2 x squared. So, first, let's set y to be equal to p of x, and our function is now y equals 12x minus 2x squared. Next, let's interchange our x values for our y values. Right? Because that's what we need to do when we're finding the inverse of a function. And now our function will be written as x equals 12 y minus y 2 y squared. Now, let's go ahead and solve for y, our new input. So, solving for y. First, you may notice that this is a quadratic equation. We have a squared term, a linear term, and a constant. So, let's rewrite it in the general form for a quadratic equation which means that I'd add 2 y squared to the left hand side, Subtract 12 y from the left hand side and I'd write back my constant x. So, 2 y squared minus 12 y plus x equals 0. Now that it's in quadratic form, I can go ahead and apply the quadratic formula. And, the quadratic formula says that given a, which is the coefficient of the squared term here would be 2. B which is the coefficient of the linear term which is negative 12, and c which is the constant would be x, then we can find our unknown value y using the formula. Let me let me write it somewhere right here. The formula tells us that our unknown equals negative b plus or minus the square root of b squared minus 4 a c all divided by 2 a. So, in order to solve, we need to rewrite our formula with our values of a, b, and c. Now, b is negative 12, so that would be negative negative 12, plus or minus the square root of b squared, negative 12 squared, minus 4 multiplied by a, which is 2, multiplied by c, which is x. And all of that is going to be divided by 2 multiplied by a, which is 2. Now, let's go ahead and simplify. Negative negative twelves are positive 12. Negative 12 squared is 144 and a negative 4 multiplied by 2 multiplied by x is negative 8 x and all of that is being divided by 4. Now, if you look under the square root, notice that we have two terms that share a common factor of 4 and 4 is a squared number. So we can rewrite what's happening under our square root as 4 multiplied by 36 minus 2 x. And all of that is being still being divided by 4. Now, why did we do that? Well, we did it because we can now apply the product rule for radicals. And recall, it says that if we're finding the square root of a product, a b is the same thing as finding the square root of a multiplied by the square root of b. So using that idea, we can now rewrite our expression as 12 plus or minus the square root of 4 multiplied by the square root of 36 minus 2 x all divided by 4, which if I come over to the right hand side of the page here, then now I can rewrite this as y equals 12 plus or minus the well, the square root of 4 is 2. So that's 2 multiplied by 36 minus 2 x. And that's all being divided by 4. If I divide through by 4, that now means y equals 12 divided by 4 which is 3, plus or minus 2 divided by 4 which is a half, multiplied by the square root of 36 minus 2 x. Now, before we implement our value because we're almost there for finding our inverse, remember that we were told p of x is decreasing. But, in our problem, we have 3 plus or minus the a half multiplied by the square root of 36 minus 2 x. So, what would we do in that function to ensure it's decreasing? Well, it couldn't be adding because that would be making it bigger. So, that means y equals 3 minus a half multiplied by the square root of 36 minus 2 x. Now, that we've solved for y, that means it's now our inverse function. So, we can rewrite it as the inverse of p of x equals 3 minus a half multiplied by the square root of 36 minus 2 x. Now that we have that, we can put in the value for the price of our particle share. Remember that value of p of x is $10. So given p of x given x equals $10 because now that would be the p of x would be the new value of x. It would be the new input. I can substitute it into my inverse function. So, the inverse of p of 10, which would be 3 minus a half, multiplied by the square root of 36 minus 2 multiplied by 10. 2 multiplied by 10 is 6 is 20, and 36 minus 20 equals 16. The square root of 16 is 4, so our problem becomes 3 minus a half of 4. A half of 4 is 2 and 3 minus 2 equals 1. So, p inverse or the inverse of p of 10 equals 1. In other words, when the price of a share is $10 and it's and p of x is decreasing, then the number of hours from 9 AM would be 1 which would have been answer choice a. Thanks a lot for watching everyone. I hope this video helped.