{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.

c. Find the inverse function that gives the time t at which the ball is at height h as the ball travels downward. Express your answer in the form t = ƒ⁻¹ (h)

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 p of x equals 12 x minus 2 x squared, where x is the number of hours from 9 AM. We want to find the inverse of p of x when the value of p of x is decreasing. Now we're looking for the inverse of p of x, which essentially is written like this. And we want to find that inverse when the value of p of x is decreasing. So Since we know that P of X equals 12 X minus 2 X squared. 1st, let's let y be equal to P of X, because that will make it easier for us to work while we go. So since that's the case, that means y equals 12 x minus 2 x squared. Now, in order for us to solve for the inverse, we can interchange Interchanging the values for X and Y. Then our equation now becomes X equals 12 y minus 2 y squared. In order for us to solve for the inverse, we need to solve for y. So solving for y, first, notice that this is a quadratic expression or a quadratic equation rather. So if we're going to solve for y, we can try to write it in its general form, which means putting all of the values on the left hand side of the equation. So to do that, we'd add 2 y squared, subtract 12 y, and leave our value of x. So our equation is now 2 y squared minus 12 y plus x equals 0. Since this is a quadratic term, we can now use the quadratic formula to solve for y. Because from our quadratic, equation, notice that a, the coefficient of the squared term is 2. B, the coefficient of the linear term is negative 12. And c, the constant equals x. So we can solve for y because now that tells us y equals negative b. So that's negative negative 12 plus or minus the square root of b squared. That's negative 12 squared minus 4 multiplied by a, which is 2, multiplied by c, which is x. And all of that is being divided by 2 multiplied by a, which is 2. So let's go ahead and continue solving. Now negative negative twelve is positive 12. We have plus or minus negative 12 squared, that's 144 and negative 4 multiplied by 2 multiplied by x is negative 8 x. And all of that is being divided by 4. Now notice that under the square root term, they share a common factor of 4. So we can factor 4 out from what's under our square root and that would know before multiplied by 36. 4 into 144 goes 36 times and 4 into negative 8 x goes negative two x times. And all of that is still being divided by 4. Now, recall, there's a property that can help us to simplify this even more because recall that for a radical or the radical of our product, the radical of a b equals the radical of a multiplied by the radical of b. In other words, we can rewrite our expression here as a square root of 4 multiplied by the square root of 36 minus 2 x. And the reason why we want to do that is because we can get a square factor from the number 4. Because remember, the square root of 4 is 2. So, If I write it, let me go over to my right hand side. No, this tells us that y equals 12 plus or minus the square root of 4, which is 2 multiplied by 36 minus 2 x. And all of that is being divided by 4. Now, we can go ahead and divide both terms by 4. 12 divided by 4 is 3. 2 divided by 4 is a half. So that's 3 plus, plus or minus sorry, a half of the square root of 36 minus 2 x. Now, we're almost there. Just remember that as we said before, the in we want the inverse when p of x is decreasing and p of x would be decreasing if we're subtracting one term from the other. So that means since p of x is decreasing, then what we really want is the difference. Therefore, y equals 3 minus a half of the square root of 36 minus 2 x. Now that we have solved for y, we can replace and y represents our inverse. That means we can rewrite y as the inverse of p of x equals 3 minus a half of the square root of 36 minus 2x. And that would have been the inverse of p of x. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Quadratic Functions

Inverse Functions

Solving Quadratic Equations

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