Recalling the Formula for the Surface Area of a Cone

Oct 13, 2025 by windailys.com 33 views

Understanding the Problem

The surface area of a cone is given as 216 square units, and we are told that the height of the cone is times greater than the radius. Our goal is to find the length of the radius of the cone to the nearest foot.

Recalling the Formula for the Surface Area of a Cone

The surface area of a cone is given by the formula:

A = πr(r + √(h^2 + r^2))

where A is the surface area, π is a mathematical constant approximately equal to 3.14, r is the radius of the base of the cone, and h is the height of the cone.

Given Information

We are given that the surface area of the cone is 216 square units, and the height of the cone is times greater than the radius. We can express this as:

h = nr

where n is a constant.

Substituting the Given Information into the Formula

Substituting the given information into the formula for the surface area of a cone, we get:

216 = πr(r + √(nr)^2 + r^2)

Simplifying the Equation

Simplifying the equation, we get:

216 = πr(r + √(n^2r2 + r^2))

Squaring Both Sides

Squaring both sides of the equation, we get:

46656 = π^2r2(r + √(n^2r2 + r²⁾⁾2

Expanding the Right-Hand Side

Expanding the right-hand side of the equation, we get:

46656 = π^2r2(r^2 + 2nr^2 + n^2r2 + r^2 + 2nr√(n^2r2 + r^2) + n^2r2 + r^2)

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

46656 = π^2r2(4n^2r2 + 4nr^2 + 4r^2)

Dividing Both Sides by π^2r2

Dividing both sides of the equation by π^2r2, we get:

(46656)/(π^2r2) = 4n^2 + 4n + 4

Simplifying the Left-Hand Side

Simplifying the left-hand side of the equation, we get:

(46656)/(π^2r2) = 4(n^2 + n + 1)

Solving for n

Solving for n, we get:

n^2 + n + 1 = (46656)/(4π^2r2)

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

n^2 + n + 1 = (11664)/(π^2r2)

Solving for r

Solving for r, we get:

r^2 = (11664)/(π²⁽ⁿ2 + n + 1))

Substituting the Value of n

Substituting the value of n, we get:

r^2 = (11664)/(π²⁽¹2 + 1 + 1))

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

r^2 = (11664)/(π^2(3))

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

r^2 = (3888)/(π^2)

Solving for r

Solving for r, we get:

r = √((3888)/(π^2))

Simplifying the Right-Hand Side

Simplifying the right-hand side of the equation, we get:

r = √(1224.6)

Solving for r

Solving for r, we get:

r = 35

Rounding to the Nearest Foot

Rounding to the nearest foot, we get:

r = 35

The final answer is: 35

Understanding the Problem

Q: What is the formula for the surface area of a cone?

A: The surface area of a cone is given by the formula:

A = πr(r + √(h^2 + r^2))

where A is the surface area, π is a mathematical constant approximately equal to 3.14, r is the radius of the base of the cone, and h is the height of the cone.

Q: What is the relationship between the height and the radius of the cone?

A: The height of the cone is times greater than the radius. We can express this as:

h = nr

where n is a constant.

Q: How do we substitute the given information into the formula for the surface area of a cone?

A: We substitute the given information into the formula for the surface area of a cone by replacing h with nr in the formula:

A = πr(r + √(nr)^2 + r^2)

Q: How do we simplify the equation?

A: We simplify the equation by combining like terms and canceling out any common factors:

216 = πr(r + √(n^2r2 + r^2))

Q: What is the next step in solving the equation?

A: The next step is to square both sides of the equation to eliminate the square root:

46656 = π^2r2(r + √(n^2r2 + r²⁾⁾2

Q: How do we expand the right-hand side of the equation?

A: We expand the right-hand side of the equation by multiplying out the terms:

46656 = π^2r2(r^2 + 2nr^2 + n^2r2 + r^2 + 2nr√(n^2r2 + r^2) + n^2r2 + r^2)

Q: How do we simplify the right-hand side of the equation?

A: We simplify the right-hand side of the equation by combining like terms:

46656 = π^2r2(4n^2r2 + 4nr^2 + 4r^2)

Q: What is the next step in solving the equation?

A: The next step is to divide both sides of the equation by π^2r2 to isolate the terms involving n:

(46656)/(π^2r2) = 4n^2 + 4n + 4

Q: How do we solve for n?

A: We solve for n by rearranging the equation to isolate n:

n^2 + n + 1 = (46656)/(4π^2r2)

Q: What is the next step in solving the equation?

A: The next step is to substitute the value of n into the equation and solve for r:

r^2 = (11664)/(π²⁽ⁿ2 + n + 1))

Q: How do we simplify the right-hand side of the equation?

A: We simplify the right-hand side of the equation by substituting the value of n:

r^2 = (11664)/(π^2(3))

Q: What is the next step in solving the equation?

A: The next step is to simplify the right-hand side of the equation:

r^2 = (3888)/(π^2)

Q: How do we solve for r?

A: We solve for r by taking the square root of both sides of the equation:

r = √((3888)/(π^2))

Q: What is the final answer?

A: The final answer is:

r = 35

The final answer is: 35