The distance of the wick to the edge of a cylindrical candle is 2⅓ inches, and the volume of wax used in the candle is approximately 159 in.?. Assuming the wick is located in the center of the candle, find the height of the candle to the nearest whole number. (1 point)
O 9 inches
• 10 inches
O 11 inches
• 12 inches
To solve this problem, we need to find the height of the candle given the distance of the wick to the edge and the volume of the wax used.
Let's denote the radius of the candle as "r" and the height as "h".
Since the wick is located in the center of the candle, we can imagine a right triangle formed by the radius, the distance of the wick to the edge, and the height.
Using the Pythagorean theorem, we can find the radius:
r^2 = (2/3)^2 + (h/2)^2
Simplifying the equation, we get:
r^2 = 4/9 + h^2/4
Multiplying through by 4 to eliminate the fraction, we have:
4r^2 = 16/9 + h^2
Multiplying by 9 to eliminate the denominator, we get:
36r^2 = 16 + 9h^2
Now, let's use the volume of the wax to find a relationship between the radius and height.
The volume of a cylinder is given by the formula:
V = πr^2h
Plugging in the given volume and radius, we have:
159 = πr^2h
Substituting the expression for r^2 from the previous equation, we get:
159 = π(16/36 + 9h^2/36)h
Simplifying, we have:
159 = (π/36)(16 + 9h^2)h
Dividing through by π/36 to isolate h, we get:
(159 * 36) / π = 16 + 9h^2
Using the value of π as approximately 3.14, we have:
(159 * 36) / 3.14 ≈ 1834.08 = 16 + 9h^2
Subtracting 16 from both sides, we get:
1834.08 - 16 ≈ 1818.08 ≈ 9h^2
Dividing through by 9, we have:
202.01 ≈ h^2
Taking the square root of both sides, we find:
h ≈ √202.01
Calculating the square root, we get:
h ≈ 14.20
Rounding the height to the nearest whole number, we have:
h ≈ 14 inches
Therefore, the height of the candle to the nearest whole number is 14 inches.