To find the dimensions of the tent that minimize the cost of the material used, we need to consider the surface area of the tent. The surface area is divided into two parts - the floor and the top/ends. Let's solve parts (a) and (b) first, where the floor is cheaper than the rest of the tent.
(a) To minimize the cost of the material used:
Step 1: Determine the dimensions of the tent.
Let's assume the side length of the equilateral triangle cross-section is "x", and the height of the prism is "h".
Step 2: Find the surface area of the floor.
The floor is an equilateral triangle, so its area is given by: Floor A = (√3 / 4) * x^2.
Step 3: Find the surface area of the top + ends.
The top and ends of the tent form 3 congruent rectangles. The width of each rectangle is "x", and the length is "h". Therefore, the total surface area of the top and ends is given by: Top + Ends A = 3 * x * h.
Step 4: Find the total surface area of the tent.
Total A = Floor A + Top + Ends A = (√3 / 4) * x^2 + 3 * x * h.
Step 5: Express the cost of the material in terms of surface area.
The cost of the floor material is "Cf" per square meter, while the cost of the top/ends material is 1.4 times more expensive, so it is 1.4 * "Cf" per square meter.
Step 6: Express the cost function.
The cost function "C" in terms of the cost of materials used can be written as:
C = Cf * Floor A + (1.4 * Cf) * Top + Ends A = Cf * (√3 / 4) * x^2 + (1.4 * Cf) * 3 * x * h.
Step 7: Express total volume and height.
The volume of the tent is given as 2.2 m^3. Since the tent is a right prism, the volume is given by: Volume = Floor A * h = (√3 / 4) * x^2 * h.
Hence, h = 2.2 / (√3 / 4) / x^2.
Step 8: Substitute the expression for "h" into the cost function.
C = Cf * (√3 / 4) * x^2 + (1.4 * Cf) * 3 * x * (2.2 / (√3 / 4) / x^2).
Simplifying the expression, we get:
C = Cf * (√3 / 4) * x^2 + (1.4 * Cf) * 6.6 / (√3 / 4).
Step 9: Differentiate C with respect to "x" to find the critical points.
Differentiating C with respect to "x", we get:
dC/dx = Cf * (√3 / 2) * x - (1.4 * Cf) * 2.2 * (√3 / 4) / x^3.
Setting this to zero, we have:
Cf * (√3 / 2) * x = (1.4 * Cf) * 2.2 * (√3 / 4) / x^3.
Simplifying, we get:
x^4 = (2.2 * 4) / (1.4 * 3) = 8 / 3.
So, x = (8 / 3)^(1/4).
(b) To find the total area of the material used:
Using the expression for the floor area and the top + ends area, we can substitute the value of "x" from part (a) into the formulas and calculate the total area.
Total A = Floor A + Top + Ends A.
Now that we've solved parts (a) and (b), let's move on to parts (c) and (d) where the floor is more expensive than the rest of the tent.
(c) To minimize the cost of the material used:
We follow a similar approach as in part (a), but this time the cost function will have different coefficients.
Step 5: The cost of the floor material is now 1.4 * "Cf" per square meter, while the cost of the top/ends material is "Cf" per square meter.
Step 6: Express the cost function.
The cost function "C" in terms of the cost of materials used can be written as:
C = (1.4 * Cf) * Floor A + Cf * Top + Ends A = (1.4 * Cf) * (√3 / 4) * x^2 + Cf * 3 * x * h.
Step 7: Express the height "h".
The volume is still 2.2 m^3, and since the height is now expressed in terms of the floor area, we have:
h = 2.2 / (√3 / 4) / x^2.
Step 8: Substitute the expression for "h" into the cost function.
C = (1.4 * Cf) * (√3 / 4) * x^2 + Cf * 3 * x * (2.2 / (√3 / 4) / x^2).
Step 9: Differentiate C with respect to "x" to find the critical points and solve for "x".
(d) To find the total area of the material used:
Using the expression for the floor area and the top + ends area, substitute the value of "x" from part (c) into the formulas and calculate the total area.