find the radius of a right cone with slant height of 21 m and surface area of 232
Surface area of 232pi
I'm sorry you don't know the difference between geometry and geography.
I meant to put geometry.
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To find the radius of a right cone with a slant height of 21 m and a surface area of 232 square units, we can use the following steps:
1. The formula to calculate the surface area of a cone is given as:
Surface Area = π * r * (r + l)
Where:
Surface Area is the given surface area of the cone
r is the radius of the base
l is the slant height
2. Rearrange the formula to solve for the radius:
Surface Area = π * r * (r + l)
Divide both sides of the equation by π:
Surface Area / π = r * (r + l)
Distribute:
Surface Area / π = r^2 + r * l
Rearrange the equation and combine like terms:
r^2 + r * l - (Surface Area / π) = 0
3. Now, we have a quadratic equation in terms of the radius, r. We can solve this equation using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula to solve for r:
The quadratic formula is given as:
r = (-b ± √(b^2 - 4ac)) / (2a)
where a, b, and c are the coefficients in the quadratic equation ax^2 + bx + c = 0
From our derived equation:
a = 1
b = l
c = -(Surface Area / π)
Substitute these values into the quadratic formula:
r = (-(l) ± √((l)^2 - 4(1)(-(Surface Area / π)))) / (2(1))
4. Simplify the equation:
r = (-l ± √(l^2 + 4(Surface Area / π))) / 2
5. Substitute the given values into the equation:
r = (-21 ± √(21^2 + 4(232) / π)) / 2
6. Calculate the value of the square root and simplify the equation:
r = (-21 ± √(441 + 928 / π)) / 2
r = (-21 ± √(441 + 2956.9158)) / 2
r = (-21 ± √(3397.9158)) / 2
r ≈ (-21 ± 58.31) / 2
7. Finally, we have two possible solutions for the radius:
r1 = (-21 + 58.31) / 2 ≈ 18.655 m
r2 = (-21 - 58.31) / 2 ≈ -39.655 m
Since a negative radius is not meaningful in this context, the radius of the cone would be approximately 18.655 m.