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.
We are given the formula for the surface area of a cone as A = πr^2 + πr√(r^2 + h^2).
First, we substitute the given values into the formula:
225 = π(4^2) + π(4)√(4^2 + h^2)
Simplifying, we get:
225 = 16π + 4π√(16 + h^2)
Dividing both sides by π, we have:
225/π = 16 + 4√(16 + h^2)
Next, let's solve for √(16 + h^2) by isolating it on one side of the equation:
√(16 + h^2) = (225/π - 16)/4
Squaring both sides of the equation and simplifying, we get:
16 + h^2 = ((225/π - 16)/4)^2
Now, let's solving for h^2 by subtracting 16 from both sides of the equation:
h^2 = ((225/π - 16)/4)^2 - 16
Taking the square root of both sides of the equation, we get:
h = √(((225/π - 16)/4)^2 - 16)
Using a calculator to evaluate the expression on the right side of the equation, we find that h ≈ 6.88 cm.
Therefore, the height of the cone is approximately 6.88 cm.
To find the height, let's start by inserting the given values into the surface area formula:
A = 𝜋r^2 + 𝜋r√(r^2 + h^2)
Now we can substitute the known values:
225 cm^2 = 𝜋(4 cm)^2 + 𝜋(4 cm)√((4 cm)^2 + h^2)
Simplifying further:
225 cm^2 = 𝜋(16 cm^2) + 𝜋(4 cm)√(16 cm^2 + h^2)
225 cm^2 = 16𝜋 cm^2 + 4𝜋 cm(√(256 cm^2 + h^2))
Now let's isolate the term containing the height:
225 cm^2 - 16𝜋 cm^2 = 4𝜋 cm(√(256 cm^2 + h^2))
Simplifying:
209 cm^2 = 4𝜋 cm(√(256 cm^2 + h^2))
To find the height, we need to isolate it. Divide both sides of the equation by 4𝜋 cm:
(209 cm^2) / (4𝜋 cm) = √(256 cm^2 + h^2)
Simplify the left side of the equation:
52.25 cm = √(256 cm^2 + h^2)
Now, square both sides of the equation to eliminate the square root:
(52.25 cm)^2 = (256 cm^2 + h^2)
Simplifying:
2730.0625 cm^2 = 256 cm^2 + h^2
To solve for h^2, subtract 256 cm^2 from both sides:
2730.0625 cm^2 - 256 cm^2 = h^2
2474.0625 cm^2 = h^2
Finally, take the square root of both sides to solve for h:
h = √(2474.0625 cm^2)
h ≈ 49.74 cm
Therefore, the height of the cone is approximately 49.74 cm.
To find the height of the cone, we can use the given information about the surface area and radius.
The formula for the surface area of a cone is:
A = πr^2 + πr√(r^2 + h^2)
We are given that the surface area (A) is 225 cm^2 and the radius (r) is 4 cm. We need to find the height (h) of the cone.
We can plug in the known values into the formula:
225 = π(4^2) + π(4)√(4^2 + h^2)
Simplifying the equation:
225 = 16π + 4π√(16 + h^2)
To isolate the term containing the height, let's subtract 16π from both sides:
225 - 16π = 4π√(16 + h^2)
Next, divide both sides of the equation by 4π:
(225 - 16π) / (4π) = √(16 + h^2)
Simplify the left side of the equation:
(225 - 16π) / (4π) ≈ 7.15 (approximately)
Now, we can square both sides to eliminate the square root:
(225 - 16π) / (4π) ≈ 7.15
[(225 - 16π) / (4π)]^2 ≈ 7.15^2
(225 - 16π)^2 / (16π)^2 ≈ 51.1225
We can solve for (225 - 16π)^2 to make the equation easier to work with:
(225 - 16π)^2 ≈ 51.1225 * (16π)^2
(225 - 16π)^2 ≈ 51.1225 * 256π^2
Now, divide both sides of the equation by 51.1225:
(225 - 16π)^2 / (51.1225 * 256π^2) ≈ 1
Solving for (225 - 16π)^2 / (51.1225 * 256π^2):
(225 - 16π)^2 / (51.1225 * 256π^2) ≈ 1
By rearranging the equation, we have:
(225 - 16π)^2 ≈ 51.1225 * 256π^2
Taking the square root to isolate (225 - 16π):
√[(225 - 16π)^2] ≈ √[51.1225 * 256π^2]
225 - 16π ≈ √[51.1225 * 256π^2]
Now, we can isolate the height (h) by subtracting the known values:
225 - 16π ≈ √[51.1225 * 256π^2] - 225
Simplifying the equation:
225 - 16π ≈ √[13067.52π^2] - 225
225 - 16π ≈ 361.451π - 225
To isolate the height (h), let's subtract 361.451π and add 225 to both sides:
16π ≈ 361.451π - 225
Adding 225:
16π + 225 ≈ 361.451π
Combine like terms:
241.5 ≈ 345.451π
Now, divide both sides by 345.451:
241.5 / 345.451 ≈ π
Simplifying the left side:
0.699 ≈ π
Therefore, the height of the cone is approximately equal to 0.699 cm.