At an athletics competition, the flight path of the discus thrown by one of the competitors can be described by the equation 𝑦 = −0.095𝑥² + 2.1𝑥 + 2.1, where 𝑦 is the height of the discus above the ground in meters when it has travelled a horizontal distance of 𝑥 meters. How far does the discus travel horizontally before it hits the ground for the first time?

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### Video Transcript

At an athletics competition, the flight path of the discus thrown by one of the competitors can be described by the equation 𝑦 equals negative 0.095𝑥 squared plus 2.1𝑥 plus 2.1, where 𝑦 is the height of the discus above the ground in metres when it has travelled a horizontal distance of 𝑥 metres. How far does the discus travel horizontally before it hits the ground for the first time?

Now, the key element is actually our equation. So we have an equation that models our flight. And that equation has 𝑦 equals negative 0.095𝑥 squared plus 2.1𝑥 plus 2.1, where 𝑦 is actually our vertical distance and 𝑥 is our horizontal distance. Well, what we want to know in this question is actually what horizontal distance, so what is the value of 𝑥, when the actual discus hits the ground for the very first time.

Well the key thing here is that actually when it hits the ground for the first time, 𝑦 is going to be equal to zero. And this is because 𝑦 is the distance above the ground. So if it’s actually gonna hit the ground, then that distance is going to be zero. So let’s substitute this into our equation. So what happens when we actually put zero into our equation is we get a quadratic. So I flipped around the other side so that we actually have zero on the right-hand side.

So now we’ve got negative 0.095𝑥 squared plus 2.1𝑥 plus 2.1 is equal to zero. And we want to solve this to find our 𝑥-values because then we’ll be able to see actually how far the discus has travelled horizontally before it’s hit the ground or got a 𝑦 value of zero. So how we’re gonna solve this? So to enable us to actually solve our quadratic, what we’re gonna use is the quadratic formula. The quadratic formula tells us that when 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero, so we have a quadratic in this form, then 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 over two 𝑎.

So then we take a look at our own quadratic that we’re trying to solve here. And we can see it’s in that form. So what we do is we actually identify our 𝑎, 𝑏, and 𝑐. So in our case, it’s gonna be 𝑎 is equal to negative 0.095. And that’s because it’s coefficient of 𝑥 squared. We’ve got 𝑏 is equal to 2.1 and 𝑐 is equal to 2.1. What you gotta do here is make sure that we are careful with signs. And that’s why I have shown it with the sign.

So we have a negative for the first coefficient. So it’s the coefficient of 𝑥 squared. And then the other two are actually positives. That’s why, as I said, I put the line underneath the sign as well as the value. Okay, so now let’s use the quadratic formula to find our value of 𝑥. So therefore, when we actually substitute in our values for 𝑎, 𝑏, and 𝑐, we get 𝑥 is equal to negative 2.1 plus or minus the square root of 2.1 all squared minus four, and then multiplied by negative 0.095 multiplied by 2.1, then all divided by two multiplied by negative 0.095.

So therefore, we’re gonna get 𝑥 is equal to negative 2.1 plus the square root 5.208 over negative 0.19. Or 𝑥 is equal to negative 2.1 minus root 5.208 over negative 0.19. So now what we’re gonna do is actually calculate those and see what the answer we get is. So when we calculate the answer, we get 𝑥 is equal to either negative 0.9584 et cetra. or 𝑥 is equal to 23.0637.

Well we can disregard the first answer because that is giving a negative answer. And if we had a negative answer, then actually that means that the flight of the discus would have to travel backwards. So that couldn’t be the case. So therefore, the distance travelled horizontally by a discus before it hits the ground for the first time when using the model 𝑦 is equal to negative 0.095𝑥 squared plus 2.1𝑥 plus 2.1 is 23.06 meters to two decimal places.