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 a cone, V=1/3pi r^2h and SA=pi r^2+pi r sqrt(r^2+h^2)
The ratio of the volume of the candle to its surface area is expressed as V/SA.
Substitute the formulas for volume and surface area of the cone into the ratio:
V/SA = (1/3πr^2h) / (πr^2 + πr√(r^2+h^2))
Simplify by factoring out πr^2:
V/SA = (1/3h) / (1 + √(r^2+h^2)/r)
Since h = r, plug in this value:
V/SA = (1/3) / (1 + √(r^2+r^2)/r)
V/SA = 1/3 / (1 + √2)
Therefore, the ratio of the volume of the candle to its surface area is 1/3 divided by (1 + √2).
the options are
a 3r-sqrt(h^2+r^2)/8r-h
b r(1-sqrt2)/-3
c rh/3r+sqrt(h^2+r^2)
d r/3+sqrt2
To determine the correct option, let's simplify the ratio we calculated and compare it to each of the provided options.
Given: V/SA = 1/3 divided by (1 + √2)
For simplicity, let's express √2 as a decimal approximation:
√2 ≈ 1.414
Plugging this into the ratio we calculated:
V/SA = 1/3 / (1 + 1.414)
V/SA = 1/3 / 2.414
V/SA ≈ 0.414
Now let's compare this value to each of the provided options:
a) 3r - √(h^2 + r^2) / 8r - h
b) r(1 - √2) / -3
c) rh / 3r + √(h^2 + r^2)
d) r / 3 + √2
From these options, the closest match to our calculated value of approximately 0.414 is option b) r(1 - √2) / -3.
Therefore, the correct option is b) r(1 - √2) / -3.