Question Video: Solving Quadratic Equations Using the Quadratic Formula in a Real-World Context

The height above the ground of a golf ball can be found using the equation ๐‘ฆ = โˆ’16.1๐‘กยฒ + 113๐‘ก + 9, where ๐‘ฆ is the height in feet and ๐‘ก 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 2 decimal places.

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Video Transcript

The height above the ground of a golf ball can be found using the equation ๐‘ฆ equals negative 16.1๐‘ก squared plus 113๐‘ก plus nine, where ๐‘ฆ is the height in feet and ๐‘ก 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 itโ€™s actually asking us. So what itโ€™s 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 ๐‘ฆ 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 ๐‘ฆ is gonna be equal to zero because ๐‘ฆ is the height in feet. So wait, we know what ๐‘ฆ is, so now we can try and solve this equation to find out what ๐‘ก 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 ๐‘ฆ is equal to zero. So now weโ€™ve got a quadratic equation that we can solve, and we can solve this to find our values for ๐‘ก. 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 weโ€™re gonna do is weโ€™re gonna to use the quadratic equation, which tells us that where ๐‘Ž๐‘ฅ squared plus ๐‘๐‘ฅ plus ๐‘ equals zero, then weโ€™ve got the quadratic equation that says that ๐‘ฅ is equal to negative ๐‘ plus or minus the square root of ๐‘ squared minus four ๐‘Ž๐‘ all over two ๐‘Ž. We have that plus or minus because actually being a quadratic, itโ€™s possible that we can have two values here.

So great, well weโ€™re gonna use this formula to solve this problem. Well, first of all, I have highlighted what are ๐‘Ž, ๐‘, and ๐‘ going to be. So ๐‘Ž is the coefficient of the ๐‘ก squared term, and ๐‘ is the coefficient of ๐‘ก, 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, weโ€™ve actually substituted in our values. So weโ€™ve got negative 113, again, itโ€™s cause itโ€™s negative ๐‘. So negative of 113 plus or minus the square root of 113 squared minus four. And then itโ€™s ๐‘Ž times ๐‘, so thatโ€™s gonna be negative 16.1 multiplied by nine, and then all divided by two multiplied by negative 16.1 which gives us ๐‘ฅ 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 weโ€™re talking about because itโ€™s 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 weโ€™re 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 canโ€™t 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 ๐‘ฅ is equal to 7.10 seconds, not forgetting our units. And weโ€™ve got 7.10 because again, if we look at our second decimal place, cause itโ€™s asking for two decimal places, weโ€™d have 7.09 in the original answer. Well, the number that comes after the nine is five or above, so itโ€™s a seven. So therefore, we round the nine up, so we round it to 10. So therefore, our final answer: 7.10 seconds.

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