WEBVTT
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Determine the volume of the given right circular cone in terms of ๐.
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To find the volume of a cone, we actually use the pyramid formula because technically a cone is a circular pyramid.
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This formula is one-third times the area of the base times the height.
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And since our base is a circle, instead of ๐ต, we can replace it with ๐๐ squared.
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So the volume of a cone is one-third times ๐ times ๐ squared, where ๐ is the radius of the circle, times the height of the cone.
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Looking at our diagram, we can see that the height is forty-eight centimeters but we do not have the radius.
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However, we can solve for the radius using this right triangle.
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We can use the Pythagorean theorem.
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Before we use the theorem, letโs see what we know about this right triangle.
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The height of this triangle is the exact same as the height of the pyramid.
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So itโs forty-eight centimeters.
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The longest side of this right triangle is gonna be sixty centimeters because this is a property of the cone.
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If the slant height is sixty on one side, then itโs also gonna be sixty on the other side.
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Now we can use the Pythagorean theorem to solve for the radius.
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The Pythagorean theorem states: The sum of the squares of the two shorter sides will be equal to the square of the longest side.
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Lets go ahead and plug these in.
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And instead of the radius equals a question mark, letโs just use ๐ฅ.
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๐ฅ squared plus forty-eight squared equals sixty squared.
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๐ฅ squared is equal to ๐ฅ squared.
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Forty-eight squared is equal to two thousand three hundred and four.
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And sixty squared is equal to three thousand six hundred.
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Now letโs subtract two thousand three hundred and four from both sides of the equation.
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So we get that ๐ฅ squared is equal to one thousand two hundred and ninety-six.
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And our final step would be to square root both sides of the equation.
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And we get that ๐ฅ equals thirty-six, which is our radius.
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Now that we know the radius is thirty-six centimeters and the height is forty-eight centimeters, we can plug it in to our volume of a cone formula.
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So now we have volume is equal to one-third times ๐ times thirty-six centimeters, which we need to square, times forty-eight centimeters.
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Letโs go ahead and square thirty-six centimeters.
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Thirty-six centimeters squared is equal to one thousand two hundred and ninety-six centimeters squared.
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Notice we couldโve made this question a little bit shorter because when we were solving for the radius, we couldโve just found the radius squared to equal one thousand two hundred and ninety-six and plug it into our formula.
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Nevertheless, we can keep going where weโre gonna be multiplying all of our numbers together.
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Now notice it said to leave our answer in terms of ๐.
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That means we donโt wanna multiply by ๐.
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We will wanna keep it in our answer as just ๐.
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So letโs go ahead and multiply our two large numbers together, and then we will multiply by the one-third.
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Which results in sixty-two thousand two hundred and eight centimeters cubed.
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This is because we multiplied centimeters squared times centimeters.
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When you multiply, you add your exponents, and a volume should be in centimeters cubed.
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So this is great.
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So now youโll take sixty-two thousand two hundred and eight and multiply it by one-third.
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Leaving your answer in terms of ๐, we get twenty thousand seven hundred and thirty-six ๐ centimeters cubed, or also it can be heard as cubic centimeters.