In a frustum of a right circular cone the diameter of the lower base is 24ft while the diameter of the upper base is 14ft if the slant height of the frustum is 13ft find the total area and the volume of the frustum.
Well, well, well, looks like we've got ourselves a frustum party! Don't worry, I'm here to clown around and help you out with some math.
First things first, let's calculate the height of the frustum. We can use the Pythagorean theorem, 'cause why not? So, h-squared equals 13 squared minus the radius difference squared.
The radius of the lower base is 24/2 = 12ft, while the radius of the upper base is 14/2 = 7ft. The difference is 12 - 7 = 5ft.
So, h-squared equals 13 squared minus 5 squared. That gives us h-squared equals 169 minus 25, which is 144. Taking the square root of both sides, we get h equals 12ft.
Now, let's calculate the volume of the frustum. It's like a big, cone-shaped puzzle, isn't it? The formula for the volume of a frustum is V equals 1/3 times pi times h times (R-squared plus r-squared plus R times r), where R is the radius of the lower base and r is the radius of the upper base.
Substituting the values, we get V equals 1/3 times pi times 12ft times (12 squared plus 7 squared plus 12 times 7).
Simplifying that expression, we get V equals 1/3 times pi times 12ft times (144 plus 49 plus 84).
Now add all those numbers together, multiply by pi, divide by 3, and voila! You'll have the volume of the frustum.
As for the total surface area, well, it's like the frustum is wearing a fancy hat! The formula for the total surface area of a frustum is A equals pi times (R plus r) times l, where l is the slant height.
Plug in the values, A equals pi times (12ft plus 7ft) times 13ft, and calculate your way to the answer!
Now, clown's honor, I'm leaving the number crunching to you. Remember, math can be fun too!
To find the total surface area and volume of the frustum of a right circular cone, we will follow these steps:
Step 1: Calculate the radii of the lower and upper bases.
The radius of the lower base (r₁) can be found by dividing the diameter (D₁) by 2.
r₁ = D₁ / 2 = 24ft / 2 = 12ft
The radius of the upper base (r₂) can be found using the same formula.
r₂ = D₂ / 2 = 14ft / 2 = 7ft
Step 2: Calculate the slant height of the full cone.
The slant height of the full cone (l) can be found using the Pythagorean theorem.
Using the radii and the given slant height of the frustum (L = 13ft) as the hypotenuse, we can find the height (h) of the full cone.
h = √(L² - (r₁ - r₂)²) = √(13² - (12 - 7)²) = √(169 - 25) = √144 = 12ft
Step 3: Calculate the heights of the frustum.
The height of the frustum (H) can be found using the similar triangles formed by the full cone and the frustum.
H = (h * (r₂ / (r₁ - r₂))) = (12ft * (7ft / (12ft - 7ft))) = (12ft * (7ft / 5ft)) = 84/5 ft = 16.8ft
Step 4: Calculate the lateral surface area, total surface area, and volume of the frustum.
The lateral surface area of the frustum (A) can be calculated using the formula:
A = π * (r₁ + r₂) * l = π * (12ft + 7ft) * 13ft = π * 19ft * 13ft ≈ 761.63 ft²
The total surface area of the frustum (T) can be calculated by adding the area of the two bases (A₁ and A₂) to the lateral surface area (A).
A₁ = π * r₁² = π * (12ft)² ≈ 452.39 ft²
A₂ = π * r₂² = π * (7ft)² ≈ 153.94 ft²
T = A + A₁ + A₂ = 761.63 ft² + 452.39 ft² + 153.94 ft² ≈ 1368.96 ft²
Finally, the volume of the frustum (V) can be calculated using the formula:
V = (1/3) * π * (r₁² + r₂² + r₁ * r₂) * H = (1/3) * π * ((12ft)² + (7ft)² + (12ft) * (7ft)) * 16.8ft ≈ 3347.56 ft³
Therefore, the total surface area of the frustum is approximately 1368.96 ft², and the volume is approximately 3347.56 ft³.
To find the total surface area and volume of the frustum, we need to break it down into different parts and then sum them up.
First, let's find the dimensions of the frustum.
The lower base has a diameter of 24ft, so its radius (r1) is half of that, which is 12ft.
The upper base has a diameter of 14ft, so its radius (r2) is half of that, which is 7ft.
The slant height (s) is given as 13ft.
Now, let's calculate the height of the frustum (h).
Using the Pythagorean theorem, we can find the height:
(h^2) = (s^2) - ((r1 - r2)^2)
Substituting the given values:
(h^2) = (13^2) - ((12 - 7)^2)
= 169 - 25
= 144
Taking the square root of both sides:
h = √144
= 12ft
Now that we have the dimensions of the frustum, let's find the total surface area and volume.
The total surface area (A) of the frustum is the sum of the areas of the lateral sides and the bases.
The lateral surface area (AL) can be calculated using the formula:
AL = π * (r1 + r2) * s
Substituting the values we found:
AL = π * (12 + 7) * 13
= π * 19 * 13
The area of the lower base (A1) is given by:
A1 = π * (r1^2)
Substituting the value of r1:
A1 = π * (12^2)
The area of the upper base (A2) is given by:
A2 = π * (r2^2)
Substituting the value of r2:
A2 = π * (7^2)
Now, we can calculate the total surface area by summing up the lateral surface area, the area of the lower base, and the area of the upper base:
A = AL + A1 + A2
The volume (V) of the frustum can be calculated using the formula:
V = (1/3) * π * h * (r1^2 + r2^2 + (r1 * r2))
Substituting the given values:
V = (1/3) * π * 12 * (12^2 + 7^2 + (12 * 7))
Now, you can substitute the values into the formulas and calculate the total surface area and volume of the frustum.
since the top diameter is 7/12 as big as the bottom diameter, the missing piece has a slant height of (7/5)(13)
So to figure the numbers for the frustrum, find them for the entire cone and subtract the value for the missing piece.
Or, since the frustrum is 7/19 as high as the whole cone, its
area is 1-(7/19)^2 of the whole cone
volume is 1-(7/19)^3 of the whole cone