Question Video: Using the Quadratic Formula to Solve Word Problems Mathematics

A shell was fired from a mortar along a trajectory described by the equation ๐‘ฆ = 0.19 + 0.31๐‘ฅ โˆ’ 0.5๐‘ฅยฒ, where ๐‘ฆ is the height of the shell above the ground in kilometres when it has travelled a horizontal distance of ๐‘ฅ kilometres. Find the horizontal distance covered by the shell before it hit the ground.

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Video Transcript

A shell was fired from a mortar along a trajectory described by the equation ๐‘ฆ equals 0.19 plus 0.31๐‘ฅ minus 0.5๐‘ฅ squared, where ๐‘ฆ is the height of the shell above the ground in kilometres when it has travelled a horizontal distance of ๐‘ฅ kilometres. Find the horizontal distance covered by the shell before it hit the ground.

So if we would fire a shell, it would go up in the air and then eventually hit the ground. And ๐‘ฆ represents the height of that shell, so when it hits the ground, ๐‘ฆ would be zero and ๐‘ฅ represents how far it would travel horizontally. So if we take our equation and we put it in descending order, we would put the ๐‘ฅ squared term first, then the ๐‘ฅ, and then the constant. So from here, we will know the distance travelled before it hit the ground and it will hit the ground when ๐‘ฆ is zero. So we said that itโ€™s equal to zero.

So we can solve this quadratic equation a few ways, but the best way since thereโ€™s lots of decimals would be the quadratic formula. The quadratic formula is negative ๐‘ plus or minus the square root of ๐‘ squared minus four ๐‘Ž๐‘ all over two ๐‘Ž. ๐‘Ž is the coefficient in front of ๐‘ฅ squared, ๐‘ is the coefficient in front of ๐‘ฅ, and ๐‘ is your constant. So taking our values, we can solve for ๐‘ฅ, the distance that it travelled horizontally, by plugging these into that quadratic formula. Now that theyโ€™ve been plugged in, letโ€™s evaluate.

So after squaring our decimal and multiplying four ๐‘Ž๐‘, letโ€™s simplify this more. And now letโ€™s take the square root and we get 0.31 plus or minus 0.69. But before we get too far, letโ€™s take a few steps back and look at our signs. The negative 0.31 cancelled with the negative with the one, so thatโ€™s why 0.31 is positive. And then since we have a plus and a minus, it wouldnโ€™t matter if we would bring the negative up with it. It would still have a plus and a minus.

So we need to take 0.31 and add 0.69 and also subtract 0.69. So we get ๐‘ฅ equals one or ๐‘ฅ equals negative 0.38. However, this is a horizontal distance, so itโ€™s not going to travel at negative amount. So that means after the shell was fired, it travelled a horizontal distance of one kilometre before it hit the ground.

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