The formula for the surface area of a cone is given by the formula 𝐴 = 𝜋𝑟2 + 𝜋𝑟√𝑟2 + ℎ2 , where A represents the surface area, r represents the radius, and h represents the height. If the surface area of a cone is 225 cm2 and the radius is 4 cm, find the height of the cone.
To find the height of the cone, we can use the given formula for the surface area and substitute the given values:
225 = π(4^2) + π(4√(4^2) + h^2)
Simplifying this equation gives:
225 = 16π + 16π + πh^2
225 = 32π + πh^2
Subtracting 32π from both sides:
225 - 32π = πh^2
Dividing both sides by π:
(225 - 32π) / π = h^2
Using a calculator to evaluate the expression on the left side:
(225 - 32π) / π ≈ 72.14
Taking the square root of both sides:
h ≈ √(72.14)
h ≈ 8.50 cm
Therefore, the height of the cone is approximately 8.50 cm.
To find the height of the cone, we can use the surface area formula and solve for h. Given that:
A = 225 cm^2 (the surface area of the cone)
r = 4 cm (the radius of the cone)
We plug these values into the formula 𝐴 = 𝜋𝑟^2 + 𝜋𝑟√𝑟^2 + ℎ^2
225 = 𝜋(4)^2 + 𝜋(4)√(4)^2 + ℎ^2
Simplifying the equation, we have:
225 = 16𝜋 + 16𝜋 + ℎ^2
225 = 32𝜋 + ℎ^2
To find the value of ℎ^2, we subtract 32𝜋 from both sides:
ℎ^2 = 225 - 32𝜋
Now, we can take the square root of both sides to find the value of h:
ℎ = √(225 - 32𝜋)
We can use a calculator to find the approximate value of ℎ.
To find the height of the cone, we can rearrange the formula for surface area and solve for h.
The formula for the surface area of a cone is: A = πr^2 + πr√(r^2 + h^2)
Given that the surface area of the cone is 225 cm^2 and the radius is 4 cm, we can substitute the values into the formula and solve for h.
225 = π(4)^2 + π(4)√(4^2 + h^2)
Simplifying this equation:
225 = 16π + 4π√(16 + h^2)
Subtracting 16π from both sides:
225 - 16π = 4π√(16 + h^2)
Dividing both sides by 4π:
(225 - 16π)/(4π) = √(16 + h^2)
Squaring both sides to remove the square root:
[(225 - 16π)/(4π)]^2 = 16 + h^2
Expanding the square on the left side:
[(225 - 16π)^2]/[(4π)^2] = 16 + h^2
Now, we can simplify the equation further:
(225 - 16π)^2 = 16(4π)^2 + (4π)^2h^2
(225 - 16π)^2 = 16(16π^2) + 16(π^2)h^2
Expanding the equation:
(225^2 - 2*225*16π + (16π)^2) = 16(16π^2) + 16π^2h^2
Now, we can simplify the equation:
50625 - 7200π + 256π^2 = 256π^2 + 16π^2h^2
Hence, the equation simplifies to:
50625 - 7200π = 16π^2h^2
Dividing both sides by 16π^2:
(50625 - 7200π)/(16π^2) = h^2
Finally, taking the square root of both sides:
h = √[(50625 - 7200π)/(16π^2)]
Using a calculator to evaluate this expression, we find that the height of the cone is approximately 5.767 cm.