Question

A candle in the shape of a circular cone has a base of radius r and a height of h that is the same length as the radius. which expresses the ratio of the volume of the candle to its surface area(including the base)? for cone, v=1/3pir^2h and sa=pir^2 pir sqrt r^2 h^2.

Answer 1

Rational Expressions QC

1.B

2.C

3.D

4.D

5.B

Answer 2

`(r(1-√(2)))/(-3)`

Explanation:

Volume of cone =
`(1)/(3) \pi r^(2) h`

Since we are given that a circular cone has a base of radius r and a height of h that is the same length as the radius

=
`(1)/(3) \pi r^(2) * r`

=
`(1)/(3) \pi r^(3)`

Surface area of cone including 1 base =
`\pi r^(2) +\pi* r * √(r^2+h^2)`

Since r = h

So, area =
`\pi r^(2) +\pi* r * √(r^2+r^2)`

=
`\pi r^(2) +\pi* r * √(2r^2)`

=
`\pi r^(2) +\pi* r^2 * √(2)`

Ratio of volume of cone to its surface area including base :

`((1)/(3) \pi r^(3))/(\pi r^(2) +\pi* r^2 * √(2))`

`((1)/(3)r)/(1+√(2))`

`(r)/(3(1+√(2)))`

Rationalizing

`(r)/(3(1+√(2))) * (1-√(2))/(1-√(2))`

`(r(1-√(2)))/(-3)`

Hence the ratio the ratio of the volume of the candle to its surface area(including the base) is
`(r(1-√(2)))/(-3)`