{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.

d. At what time is the ball at a height of 30 ft on the way up?

d. At what time is the ball at a height of 30 ft on the way up?

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 value of x when the value of p of x is $16 and the p of x is increasing. For our answer choices, a is 2, b is 3, c is 4, and d is 5. Now what do we really know here? Well, we want to find the value of x when the value of p of x is $16. Let's think a little bit about what p of x is telling us. For our function p, what it's really saying is that we are given the number of hours from 9 AM. So we're given the hours and then our function will tell us what the price will be. But in our problem, we already know the price. Now, what we really want is once given the price, then we can find our hours. So since the function p takes the hours and gives the price, then we need an inverse function that given the price, it can give us the hours. So we need to find the inverse of p of x and then substitute the price that we know, the $16, to figure out how much hours it would be from 9 AM. In other words, the value of x. So let's go ahead and do that. So For the function, P of X equals 12 X minus 2 X squared. Let's let y be equal to P of X. So that tells us that y equals 12 x minus 2 x squared. And it will be easier to manipulate as we go along. Let's also interchange the x's for y's. In our equation. And now, that becomes x equals 12 y minus 2 y squared. So now, we need to solve for y and then rewrite it as our inverse function. Notice that this is a quadratic. So let's make the right hand side of the equation 0. So by doing that, I can add 2 y to the left hand side. 2 y squared. Subtract 12 y and I put back my x. So 2 y squared minus 12 y plus x equals 0. Now it's in the form of a quadratic. So I can go ahead and use the quadratic formula. So given a equals 2, b equals negative 12, the coefficient of a linear term and our constant equals x. Okay. Then by using the quadratic formula to solve. Okay. That tells us why our unknown value equals negative negative 12 plus or minus the square root of 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. Now negative negative 12 becomes positive 12. Negative 12 squared is 144. 4 multi negative 4 multiplied by 2 is negative 8. So that's negative 8 x. And all of that is being divided by 4. Now notice that the terms under the square root have a common factor of 4. So we can factor 4 out to write 12 plus or minus the square root of 4 multiplied by 36 minus 2 x. All of that is still divided by 4. We can take it a step further because based on recall that based on what we know about radicals, then we know that the radical a b, the radical of our product equals the radical of a multiplied by the radical of b. So we can. Oops. My bad. I need a rubber. We can rewrite our expression under the square root as a square root of 4 multiplied by the square root of 36 minus 2 x. And all of that is still being divided by 4. Let me make some space over to the right here. And now, that means y equals 12 plus or minus the square root of 4 is 2 And that is still being multiplied by 36 minus 2 x all divided by 4. No notice that each term has a common factor of 2. So by factoring 2 out, we can have 12 divided by 2, which is 6 and 2 divided by 2. That's 1. So that would be 6 plus or minus the square root of 36 minus 2 x. And all of that is being divided by 2. Because when you factor 2 out of 4, we get 2. Now, before we continue, remember that we said the value of p of x is increasing. So that means we'll be working with a positive value. If we were to subtract our expression under the square root, then it would more likely be negative than if we were to add it. So that means y is actually equal to 6 plus the square root of 36 minus 2 x, all divided by 2. And know that we solved for y, we can replace it because y now represents the inverse function. So replacing y with p inverse of x, this tells us that our inverse function equals 6 plus the square root of 36 minus 2 X all divided by 2. In other words, this is the function. This is the function that is going to give us the number of hours given the price of the share. So now that we're at this point, all we need to do now is to substitute 16 into our formula. So now this tells us that p inverse of 16, which would be the value of x. So let me write that part first. The value of x or the number of rsp inverse 16 equals 6 plus the square root of 36 minus 2 multiplied by x, which is 16 and that's all being divided by 2. Now 2 multiplied by 16 is 32 and 36 minus 32 equals 4. We can find the square root of 4 because the square root of 4 is 2. So our function is equal to 6 +2, all divided by 2 and 6 plus 2 equals 8. 8 divided by 2 equals 4. So this tells us therefore that when p of x equals 16, in other words, $16, in other words, when the price of a share is $16, then the number of hours x equals 4. And when we go back to our answer choices, that would have been answer choice c. Thanks a lot for watching everyone. I hope this video helped.

{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.

d. At what time is the ball at a height of 30 ft on the way up?

Key Concepts

Quadratic Functions

Solving Quadratic Equations

Interpreting Function Values

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