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A shell was fired from a mortar along a trajectory described by the equation ๐ฆ equals 0.19 plus 0.31๐ฅ minus 0.5๐ฅ squared, where ๐ฆ is the height of the shell above the ground in kilometres when it has travelled a horizontal distance of ๐ฅ kilometres.
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Find the horizontal distance covered by the shell before it hit the ground.
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So if we would fire a shell, it would go up in the air and then eventually hit the ground.
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And ๐ฆ represents the height of that shell, so when it hits the ground, ๐ฆ would be zero and ๐ฅ represents how far it would travel horizontally.
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So if we take our equation and we put it in descending order, we would put the ๐ฅ squared term first, then the ๐ฅ, and then the constant.
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So from here, we will know the distance travelled before it hit the ground and it will hit the ground when ๐ฆ is zero.
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So we said that itโs equal to zero.
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So we can solve this quadratic equation a few ways, but the best way since thereโs lots of decimals would be the quadratic formula.
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The quadratic formula is negative ๐ plus or minus the square root of ๐ squared minus four ๐๐ all over two ๐.
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๐ is the coefficient in front of ๐ฅ squared, ๐ is the coefficient in front of ๐ฅ, and ๐ is your constant.
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So taking our values, we can solve for ๐ฅ, the distance that it travelled horizontally, by plugging these into that quadratic formula.
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Now that theyโve been plugged in, letโs evaluate.
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So after squaring our decimal and multiplying four ๐๐, letโs simplify this more.
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And now letโs take the square root and we get 0.31 plus or minus 0.69.
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But before we get too far, letโs take a few steps back and look at our signs.
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The negative 0.31 cancelled with the negative with the one, so thatโs why 0.31 is positive.
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And then since we have a plus and a minus, it wouldnโt matter if we would bring the negative up with it.
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It would still have a plus and a minus.
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So we need to take 0.31 and add 0.69 and also subtract 0.69.
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So we get ๐ฅ equals one or ๐ฅ equals negative 0.38.
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However, this is a horizontal distance, so itโs not going to travel at negative amount.
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So that means after the shell was fired, it travelled a horizontal distance of one kilometre before it hit the ground.