An architect is designing an A-frame cottage for a resort. A cross section of the cottage is an isosceles triangle. The front wall of the cottage must accommodate a sliding door.
Knowing that the architect is allowed a rectangular space of 240 m³ in volume to build inside, and the cottage should be 10 m long, determine the width and the height of the cottage that minimizes the cost of the building material, then determine the dimentions of the door with the maximum area.
To determine the width and height of the cottage that minimizes the cost of the building material, we need to set up an equation and find its minimum value.
Let's assume the width of the base of the triangular cottage is 'x' meters, and the height is 'h' meters.
The volume of the triangular cottage is equal to the volume of the rectangular space, so we have:
Volume = Base Area * Height
240 = (1/2 * x) * h
Simplifying the equation:
480 = x * h
To minimize the cost of the building material, we need to minimize the surface area of the triangular cottage.
The surface area of the triangular cottage consists of the front wall, the two side walls, and the roof. We can find the surface area using the formula:
Surface Area = Base Perimeter * Height/2 + Base * Roof Height
Perimeter of the base (triangle) = 2x + 10 (10m is the length of the cottage)
Now, let's substitute the value of h from the volume equation into the surface area equation to get it in terms of a single variable:
Surface Area = (2x + 10) * (480/x) / 2 + x * Roof Height
Surface Area = (2x + 10) * (480/x) / 2 + x * Roof Height
To minimize the cost of building material, find the derivative of the surface area with respect to x and set it equal to zero:
d(Surface Area)/dx = (x^2 - 960/x^2) + Roof Height = 0
Multiplying through by x^2:
x^4 - 960 + x^2 * Roof Height = 0
Now, you need to know the value of Roof Height to solve the equation. Is the roof a fixed height or does it have a specific slope?
To determine the width and height of the cottage that minimizes the cost of building material, we need to first establish a relationship between the volume, the dimensions, and the cost of building material.
Given that the volume of the cottage is 240 m³, and we have an isosceles triangle as the cross-section, we can derive the volume formula for the cottage.
Volume of a triangular prism = (1/2) * base * height * length
Since the triangle is isosceles, and the length is given as 10 m, we can denote the base of the triangle as b and the height as h. Therefore, the volume equation becomes:
240 = (1/2) * b * h * 10
480 = b * h
Now, we need to minimize the cost of building material. Let's assume the cost of material per unit area of the triangular cross-section is C.
The cost function can be represented as:
Cost = C * (base + 2 * height) * 10
Cost = 10C * (b + 2h)
To minimize the cost, we need to minimize the cost function with respect to b and h while considering the constraint 480 = b * h.
To solve the problem, we can use the method of Lagrange multipliers:
1. Set up the Lagrangian:
L = 10C * (b + 2h) + λ(480 - bh)
2. Calculate the partial derivatives with respect to b, h, and λ, and set them equal to zero:
∂L/∂b = 10C - λh = 0
∂L/∂h = 20C - λb = 0
∂L/∂λ = 480 - bh = 0
3. Solve the system of equations to find the values of b, h, and λ.
From the first equation: 10C = λh
From the second equation: 20C = λb
Dividing these two equations: 20C/10C = λb/λh
2 = b/h
Substituting this value into the constraint equation: 480 = b * h
480 = 2h^2
240 = h^2
h = √240 ≈ 15.49 m
Using the relationship b = 2h, we find:
b = 2 * 15.49 ≈ 30.98 m
Therefore, the width (base) of the cottage that minimizes the cost of the building material is approximately 30.98 m, and the height is approximately 15.49 m.
To determine the dimensions of the door with the maximum area, we need to find the height of the triangle that corresponds to the maximum area. Since the triangle is isosceles, the height coincides with the height of the cottage, which is approximately 15.49 m.
Now, for the dimensions of the door, it depends on the desired proportion. If we assume the sliding door should be a square, then the dimensions would be 15.49 m by 15.49 m. However, if we assume the door should be rectangular, we would need additional information to determine the specific dimensions.