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 first need to solve the given equation 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 is 225 cm^2 and the radius is 4 cm, we can substitute these values into the equation:
225 = π(4^2) + π(4)√(4^2 + h^2)
Simplifying,
225 = 16π + 4π√(16 + h^2)
Dividing both sides of the equation by π,
225/π = 16 + 4√(16 + h^2)
This can be rewritten as,
225/π - 16 = 4√(16 + h^2)
Squaring both sides of the equation,
(225/π - 16)^2 = 4^2(16 + h^2)
Expanding and simplifying,
(225^2/π^2 - 2*225*16/π + 16^2) = 16(16 + h^2)
Multiply both sides of the equation by π^2 to eliminate the fraction:
225^2 - 2*225*16π + (16π)^2 = 16π^2(16 + h^2)
Simplifying further,
225^2 - 2*225*16π + 16^2π^2 = 256π^2 + 16π^2h^2
Rearranging the equation,
h^2 = (225^2 - 2*225*16π + 16^2π^2 - 256π^2) / (16π^2)
Calculating the values,
h^2 = (225^2 - 2*225*16*3.14159 + 16^2*3.14159^2 - 256*3.14159^2) / (16*3.14159^2)
h^2 = (50625 - 7234.56 + 804.25 - 2570.88) / 804.25
h^2 = (41024.81) / 804.25
h^2 = 51.02
Taking the square root of both sides,
h ≈ √51.02
h ≈ 7.14
Therefore, the height of the cone is approximately 7.14 cm.
The formula for the surface area of a cone is given by 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 these values into the formula.
225 = 𝜋 * (4)^2 + 𝜋 * 4 * √(4^2 + ℎ^2).
225 = 16𝜋 + 4𝜋 * √(16 + ℎ^2).
225 = 16𝜋 + 4𝜋 * √(256 + ℎ^2).
225 = 16𝜋 + 4𝜋 * (16 + ℎ^2).
225 = 16𝜋 + 64𝜋 + 4𝜋ℎ^2.
225 = 80𝜋 + 4𝜋ℎ^2.
4𝜋ℎ^2 = 225 - 80𝜋.
𝜋ℎ^2 = (225 - 80𝜋)/4.
𝜋ℎ^2 = (225/4) - 20𝜋.
𝜋ℎ^2 = 56.25 - 20𝜋.
𝜋ℎ^2 = 56.25 - (20 * 3.14).
𝜋ℎ^2 = 56.25 - 62.80.
𝜋ℎ^2 = -6.55.
Since the area cannot be negative, this result is not valid.
Therefore, there is no solution for the height of the cone with the given parameters.
To find the height of the cone, we need to use the given surface area (A) and radius (r) in the formula:
A = πr^2 + πr√(r^2 + h^2)
We are given that the surface area (A) is 225 cm² and the radius (r) is 4 cm. Plugging these values into the formula, we have:
225 = π(4^2) + π(4)√(4^2 + h^2)
225 = 16π + 4π√(16 + h^2)
To proceed with finding the height (h), we need to isolate it on one side of the equation.
225 - 16π = 4π√(16 + h^2)
Now, we square both sides of the equation to eliminate the square root:
(225 - 16π)^2 = (4π√(16 + h^2))^2
(225 - 16π)^2 = 16π^2(16 + h^2)
Expanding and simplifying both sides of the equation gives:
225^2 - 2 × 225 × 16π + (16π)^2 = 256π^2 + 16π^2h^2
Now we can solve for h by rearranging the equation:
256π^2 + 16π^2h^2 = 225^2 - 2 × 225 × 16π + (16π)^2
256π^2 + 16π^2h^2 = 50625 - 7200π + 256π^2
16π^2h^2 = 50625 - 7200π
Dividing through by 16π^2 gives:
h^2 = (50625 - 7200π) / (16π^2)
Now, take the square root of both sides to find h:
h = √ [ (50625 - 7200π) / (16π^2) ]
Using a calculator, we can find the approximate value of h.
The formula for the surface area of a cone is given by the
formula 𝐴 = 𝜋𝑟
2 + 𝜋𝑟√𝑟
2 + ℎ
2 , 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.
Given:
A = 225 cm²
r = 4 cm
We are asked to find the height of the cone (h).
Using the given values, we can substitute them into the formula for surface area and solve for h:
A = 𝜋r^2 + 𝜋r√(r^2 + h^2)
225 = 𝜋(4^2) + 𝜋(4)√(4^2 + h^2)
225 = 16𝜋 + 4𝜋√(16 + h^2)
225 - 16𝜋 = 4𝜋√(16 + h^2)
To isolate the square root term, we need to divide both sides of the equation by 4𝜋:
(225 - 16𝜋) / (4𝜋) = √(16 + h^2)
Simplifying the left side gives:
56.25 - 4π = √(16 + h^2)
Now, square both sides of the equation to eliminate the square root:
(56.25 - 4π)^2 = (16 + h^2)
Expanding and simplifying the left side gives:
3164.0625 - 450π + 16π^2 = 16 + h^2
Rearranging the equation gives:
h^2 = 3164.0625 - 450π + 16π^2 - 16
h^2 = 3164.0625 - 450π + 16(π^2 - 1)
h^2 = 3164.0625 - 450π + 16(π + 1)(π - 1)
Using a calculator to find the values of the constants, we get:
h^2 ≈ 3164.0625 - 450(3.1416) + 16(3.1416 + 1)(3.1416 - 1)
h^2 ≈ 3164.0625 - 1413.7164 + 16(4.1416)(2.1416)
h^2 ≈ 3164.0625 - 1413.7164 + 138.280704
h^2 ≈ 1888.626804
Taking the square root of both sides gives:
h ≈ √(1888.626804)
h ≈ 43.424 cm
Therefore, the height of the cone is approximately 43.424 cm.