The surface area of the ice cream cone shown below is given by A = 𝜋r2 +𝜋rs, where r is the radius of the circular top of the ice cream cone and s is the slant height of the cone. If the area of the cone is 12.16π in2 and the slant height of the cone is 6 in., find the radius of the cone. (Round your answer to one decimal place.)
To find the radius of the cone, we can use the given formula for the surface area of the ice cream cone: A = 𝜋r^2 + 𝜋rs. We are given that the surface area of the ice cream cone is 12.16π in^2.
We are also given that the slant height of the cone is 6 inches, denoted by s = 6.
Using the formula for the surface area, we can substitute the given values:
12.16π = 𝜋r^2 + 𝜋(6)(r)
To find the radius, we need to solve this equation. Let's simplify it:
12.16π = 𝜋r^2 + 6𝜋r
Now, let's isolate the terms with r on one side:
12.16π - 6𝜋r = 𝜋r^2
Next, let's rearrange the equation to make it a quadratic equation, by moving all terms to one side:
𝜋r^2 + 6𝜋r - 12.16π = 0
Now, we have a quadratic equation of the form ar^2 + br + c = 0, where a = 𝜋, b = 6𝜋, and c = -12.16π.
Since this quadratic equation does not factor easily, we can use the quadratic formula to find the values of r:
r = (-b ± √(b^2 - 4ac)) / (2a)
Let's substitute the values into the quadratic formula:
r = (-6𝜋 ± √((6𝜋)^2 - 4(𝜋)(-12.16π))) / (2𝜋)
Now, let's simplify this expression:
r = (-6𝜋 ± √(36𝜋^2 + 48.64𝜋^2)) / (2𝜋)
r = (-6𝜋 ± √(84.64𝜋^2)) / (2𝜋)
r = (-6𝜋 ± 9.2𝜋) / (2𝜋)
Simplifying further, we have:
r = -3 ± 4.6
Now, we can consider both cases separately:
1. If r = -3 + 4.6 = 1.6, we have a positive radius.
2. If r = -3 - 4.6 = -7.6, we have a negative radius, which is not meaningful in this context.
Since the radius of a cone cannot be negative, we discard the solution r = -7.6.
Therefore, the radius of the cone is approximately 1.6 inches (rounded to one decimal place).
really? Just plug in your numbers
πr^2 + πrs = A
r^2 + 6r - 12.16 = 0
(r+7.6)(r-1.6) = 0
r = 1.6