The highway department keeps its sand in a conical storage building that is 24 feet high and 64 feet in diameter. To estimate the cost of painting the building, the lateral surface area of the cone is needed. To the nearest square foot, what is the area?
The lateral surface = πrs, where s = the diagonal distance from base to top, r = radius.
s^2 = h^2 + r^2 (Pythagorean theorem)
I'll let you do the calculations.
Ah, painting a cone-shaped building, huh? Quite the task! Well, to find the lateral surface area of a cone, we can use the formula πrl, where r is the radius and l is the slant height.
Now, since the diameter is given (64 feet), we can find the radius by dividing it by 2. So, the radius (r) would be 32 feet, and the height (h) would be 24 feet.
To find the slant height (l), we can use the Pythagorean theorem, with the height (h) being one leg and the radius (r) being the other. So, using a bit of mathematical magic, we find that the slant height (l) is approximately 36.06 feet (rounded to two decimal places).
Now, we can finally use the formula! With π being approximately 3.14, the lateral surface area of the cone would be:
πrl = 3.14 * 32 feet * 36.06 feet ≈ 3631 square feet (rounded to the nearest whole number).
So, to estimate the cost of painting the building, we would need approximately 3631 square feet of paint. Let's hope they have a clown-themed paint job in mind! 🤡
To find the lateral surface area of the cone, we need to calculate the curved surface area of the cone's side.
The formula for the lateral surface area of a cone is given by: Lateral Surface Area = π × r × l, where r is the radius of the base and l is the slant height of the cone.
To calculate the slant height (l) of the cone, we can use the Pythagorean theorem. The slant height (l) is the hypotenuse of a right triangle formed by the height (h) and the radius (r) of the cone.
Given:
Height (h) = 24 feet
Diameter (d) = 64 feet
First, let's calculate the radius (r) of the cone:
Radius (r) = Diameter (d) / 2 = 64 feet / 2 = 32 feet
Next, let's calculate the slant height (l) of the cone using the Pythagorean theorem:
l² = r² + h²
l² = 32² + 24²
l² = 1024 + 576
l² = 1600
l = √1600
l = 40 feet (rounded to the nearest foot)
Now, we can calculate the lateral surface area of the cone:
Lateral Surface Area = π × r × l
Lateral Surface Area = π × 32 feet × 40 feet
Lateral Surface Area ≈ 402.12385 square feet (rounded to the nearest square foot)
Therefore, the approximate lateral surface area of the cone is 402 square feet.
To find the lateral surface area of a cone, we need to find the slant height (l) and the circumference of the base (C).
First, let's find the slant height:
Since we are given the height (h) and the diameter (d) of the cone, we can use the Pythagorean theorem to find the slant height.
The formula for the slant height of a cone is:
l = sqrt(r^2 + h^2)
where r is the radius.
Given that the diameter is 64 feet, we can find the radius (r) by dividing the diameter by 2:
r = d/2 = 64/2 = 32 feet
Now we can find the slant height:
l = sqrt(32^2 + 24^2)
l = sqrt(1024 + 576)
l = sqrt(1600)
l = 40 feet
Next, let's find the circumference of the base (C):
The formula for the circumference of a circle is:
C = 2πr
where π is a constant approximately equal to 3.14159.
Given that the radius (r) is 32 feet, we can find the circumference:
C = 2π(32)
C ≈ 6.28318(32)
C ≈ 200.96 feet
Finally, let's calculate the lateral surface area (A) of the cone:
The formula for the lateral surface area of a cone is:
A = πrl
Given that the slant height (l) is 40 feet, we can find the lateral surface area:
A = 3.14159(32)(40)
A ≈ 40212.48 square feet
Therefore, to the nearest square foot, the lateral surface area of the cone is approximately 40212 square feet.