Video Transcript
The height above the ground of a golf ball can be found using the equation ๐ฆ equals negative 16.1๐ก squared plus 113๐ก plus nine, where ๐ฆ is the height in feet and ๐ก is the time in seconds after the ball was struck. How long will it take for the ball to reach the ground? Give your answer to two decimal places.
To be able to solve this problem, we first have to think about what itโs actually asking us. So what itโs asking is, it says how long will it take for the ball to reach the ground. Well, this is really useful, because what does the ball hitting the ground actually tell us. Well, this can actually tell us what ๐ฆ is because of it saying that we want to know how long it will take for the ball to reach the ground. Well, when the ball has reached the ground, then ๐ฆ is gonna be equal to zero because ๐ฆ is the height in feet. So wait, we know what ๐ฆ is, so now we can try and solve this equation to find out what ๐ก is, so how long it will take for the ball to reach the ground.
So now looking at the equation, first we wanna do is substitute that ๐ฆ is equal to zero. So now weโve got a quadratic equation that we can solve, and we can solve this to find our values for ๐ก. Now look at the equation we have here, we can see that actually, yeah, will be very difficult to factor as something like this because of the values that we have. So what weโre gonna do is weโre gonna to use the quadratic equation, which tells us that where ๐๐ฅ squared plus ๐๐ฅ plus ๐ equals zero, then weโve got the quadratic equation that says that ๐ฅ is equal to negative ๐ plus or minus the square root of ๐ squared minus four ๐๐ all over two ๐. We have that plus or minus because actually being a quadratic, itโs possible that we can have two values here.
So great, well weโre gonna use this formula to solve this problem. Well, first of all, I have highlighted what are ๐, ๐, and ๐ going to be. So ๐ is the coefficient of the ๐ก squared term, and ๐ is the coefficient of ๐ก, and then ๐ is our final integer. And looking at these numbers, what we can actually see is that, be careful and remember to include the signs. So for instance, the first one is negative 16.1 because that will come very important when we try to work this out.
Okay, so now, weโve actually substituted in our values. So weโve got negative 113, again, itโs cause itโs negative ๐. So negative of 113 plus or minus the square root of 113 squared minus four. And then itโs ๐ times ๐, so thatโs gonna be negative 16.1 multiplied by nine, and then all divided by two multiplied by negative 16.1 which gives us ๐ฅ is equal to negative 113 plus the square root of 13348.6 over negative 32.2 or negative 113 minus square root of 13348.6 over negative 32.2. And those are the two values weโre talking about because itโs a quadratic.
Now we can solve those using a calculator. When we work that out, we actually get two values like we have mentioned. So we get 7.0973957 or negative 0.0787621, et cetera. But we know that we can actually disregard one of these values because weโre talking about how long will it take for the ball to reach the ground, and we know that the negative value will not be the correct one because it canโt be a negative time. So therefore, our answer is 7.0973957. However, is this the final answer? And this is again, be very very careful cause this is where people lose marks. Always read the question, like I said, very careful, read the question: give your answer to two decimal places.
So we get our final answer ๐ฅ is equal to 7.10 seconds, not forgetting our units. And weโve got 7.10 because again, if we look at our second decimal place, cause itโs asking for two decimal places, weโd have 7.09 in the original answer. Well, the number that comes after the nine is five or above, so itโs a seven. So therefore, we round the nine up, so we round it to 10. So therefore, our final answer: 7.10 seconds.
