Divide until the degree of the divisor is greater than the degree of the dividend.
1/3+2/18r+3
Practice makes perfect
We want to calculate what fraction of a cylindrical can is empty when filled with three tennis balls. In order to do so, we will calculate the volume of each of these objects separately.
Tennis Balls
Assuming that the tennis balls are spherical, we can use the formula for the volume of a sphere to calculate the volume of each tennis ball.
V=4/3Ï€ r^3
Since we have three tennis balls we will multiply this expression by 3.
4/3π r^3 * 3 ⇒ 4π r^3The radius of each ball is given by r. Therefore, the volume occupied by the three tennis balls is 4π r^3.
Cylindrical Can
We can use the formula for the volume of a cylinder to calculate the volume of the can.
V=Ï€ r^2 h
We are told that the radius of this can is r and its height is 6r+1. Let's substitute the expression for the height into the formula to find the volume of the can.
We can find how much of the can is empty by subtracting the volume occupied by the tennis balls from the volume of the can.
6Ï€ r^3+Ï€ r^2 - 4Ï€ r^3 = 2Ï€ r^3+Ï€ r^2
In order to find what fraction of the can is empty, we divide the empty portion of the can by the volume of the can.
We can simplify this expression using polynomial long division. In order to do so, all the terms of the dividend must be present and the polynomial must be in standard form. Since there are no missing terms and our polynomial is in descending degree order, we do not need to rewrite the polynomial. Let's divide!
l r 6 r^3 + r^2 & |l 2 r^3 + r^2
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Divide
2 r^3/6 r^3= 1/3
r 1/3 r 6 r^3 + r^2 & |l 2 r^3 + r^2
Multiply 1/3 by 6r^3+r^2
r 1/3 rl 6r^3+r^2 & |l 2r^3+r^2 & 2r^3+ 13r^2
Subtract down
r 1/3 r 6 r^3 + r^2 & |l 23 r^2
The quotient is 13 with a remainder of 23 r^2. Let's write this in the requested form.
Quotient+Remainder/Divisor ⇓
1/3+23r^2/6r^3+r^2
We can rewrite the last term of this expression.
