1. Find the expense at 60 cents a square foot of polishing the curved surface of a marble column in the shape of the frustum of a right circular cone whose slant height is 12 ft. and the radii of whose bases are 3ft 6in and 2ft 4in respectively.

Well, polishing marble is quite the fancy job! Let's calculate the area of the curved surface first.

The formula to find the area of the curved surface of a frustum of a right circular cone is A = π * (r₁ + r₂) * l, where r₁ is the radius of the larger base, r₂ is the radius of the smaller base, and l is the slant height.

Converting the measurements to feet, we have r₁ = 3.5 ft, r₂ = 2.333 ft, and l = 12 ft.

Now, let's plug them into the formula:
A = π * (3.5 + 2.333) * 12
A ≈ 3.14 * 5.833 * 12
A ≈ 217.42 square feet

Nice! The area of the curved surface is approximately 217.42 square feet.

Now, since you mentioned the cost per square foot is 60 cents, we can calculate the expense.
Expense = Area * Cost per square foot

Expense = 217.42 * 0.60
Expense ≈ $130.45

So, polishing the curved surface of the marble column would cost approximately $130.45 at 60 cents per square foot.

Hope that helps, and may your marble column shine brighter than a disco ball!

To find the expense of polishing the curved surface of the marble column, we first need to calculate the surface area of the frustum of the right circular cone.

Step 1: Convert the given radii to feet.
The radius of the larger base is 3 ft 6 in, which is equal to 3.5 ft.
The radius of the smaller base is 2 ft 4 in, which is equal to 2.333 ft (since 1 ft = 12 in, 4 in = 4/12 ft = 0.333 ft).

Step 2: Calculate the slant height.
The given slant height is 12 ft.

Step 3: Calculate the lateral surface area of the frustum of the cone.
The lateral surface area is given by the formula: A = π(R + r)l, where R and r are the radii of the bases, and l is the slant height.

Plugging in the values, we have:
A = π(3.5 + 2.333) * 12
A ≈ 135.293 ft²

Step 4: Calculate the expense.
The expense is given at 60 cents per square foot.
So, the total expense would be:
Total Expense = Surface Area * Expense per square foot
Total Expense = 135.293 * $0.60
Total Expense ≈ $81.18

Therefore, the expense of polishing the curved surface of the marble column would be approximately $81.18.

To find the expense of polishing the curved surface of the marble column, we need to calculate the surface area of the frustum of the right circular cone and then multiply it by the cost per square foot.

1. First, let's calculate the height of the frustum of the cone. Since the slant height is given as 12 ft, we can use the Pythagorean theorem to find the height (h).
- Let's represent the height of the frustum as 'h' and the slant height as 'l'.
- The radius of the larger base is 3 ft 6 in, which can be converted to 3.5 ft, and the radius of the smaller base is 2 ft 4 in, which can be converted to 2.33 ft.
- Using the Pythagorean theorem, we have: (radius of larger base - radius of smaller base)^2 + h^2 = l^2
- (3.5 - 2.33)^2 + h^2 = 12^2
- 1.17^2 + h^2 = 144
- 1.3689 + h^2 = 144
- h^2 = 144 - 1.3689
- h^2 = 142.6311
- h = √142.6311
- h ≈ 11.94 ft

2. Now, let's calculate the surface area of the frustum of the cone.
- The surface area of a frustum of a cone is given by the formula: π(R + r)l, where R is the radius of the larger base, r is the radius of the smaller base, and l is the slant height.
- π(3.5 + 2.33) * 12 ≈ 180.68 sq ft

3. Finally, to find the expense of polishing the curved surface, we multiply the surface area by the cost per square foot, which is given as 60 cents.
- Expense = 180.68 * $0.60
- Expense ≈ $108.41

Therefore, the expense at 60 cents per square foot of polishing the curved surface of the marble column would be approximately $108.41.