Imagine making a tent in the shape of a right prism whose cross-section is an equilateral triangle (the door is on one of the triangular ends). Assume we want the volume to be 2.2 m3, to sleep two or three people. The floor of the tent is cheaper material than the rest: assume that the material making up the ends and the top of the tent is 1.4 times as expensive per square meter as the material touching the ground.
(a) What should the dimensions of the tent be so that the cost of the material used is a minimum?
(b) What is the total area of the material used?
If the tent ends have side s and the tent has length t, the area is
a = 2(√3/4 s^2) + 3st
since the volume of the tent is 2.2,
√3/4 s^2 t = 2.2, and so
t = 8.8/(√3 s^2)
If the cost of the floor is 1, then the cost of the whole tent is
c = 2(√3/4 s^2)(1.4) + 2st + 1.4st
= 0.7√3 s^2 + 3.4s(8.8/(√3 s^2))
= 1.212 s^2 + 17.274/s
dc/ds = 2.414s - 17.274/s^2
= (2.414s^3 - 17.274)/s^2
dc/ds=0 when s=1.927
so, t=1.368
To find the dimensions that minimize the cost of the material used for the tent, we need to consider the surface area and volume of the tent, as well as the cost per square meter for each part.
Let's start by finding the dimensions of the tent.
Step 1: Determine the dimensions of the tent:
Let's assume the height of the tent is "h" and the base length of the equilateral triangle is "s". The side length of the equilateral triangle can be calculated using the height and the base length as follows:
Using the Pythagorean theorem, the height of the equilateral triangle can be found:
h^2 = s^2 - (s/2)^2
h^2 = s^2 - s^2/4
h^2 = 3s^2/4
h = sqrt(3s^2/4)
h = s * sqrt(3)/2
Step 2: Find the volume of the tent:
The volume of a prism is given by V = base area * height.
Since the base of the tent is an equilateral triangle, its area can be calculated using the formula: base area = (sqrt(3) / 4) * s^2.
Using the given volume, we can set up the equation:
2.2 = (sqrt(3) / 4) * s^2 * (s * sqrt(3) / 2)
2.2 = (sqrt(3)^2 / 8) * s^3
s^3 = 2.2 * 8 / (sqrt(3)^2)
s^3 = 17.6 / 3
s = (17.6 / 3)^(1/3)
Step 3: Find the height of the tent:
Substituting the value of s into the equation for the height we derived earlier:
h = (17.6 / 3)^(1/3) * sqrt(3) / 2
(a) So, the dimensions of the tent that minimize the cost of the material used are s = (17.6 / 3)^(1/3) and h = (17.6 / 3)^(1/3) * sqrt(3) / 2.
Now let's calculate the total area of the material used.
Step 4: Find the surface area of the tent:
The surface area of the tent can be calculated by adding the areas of the two ends (equilateral triangles) and the lateral surface area (rectangle).
Ends area = 2 * (sqrt(3) / 4) * s^2
Lateral surface area = s * h
Total surface area = Ends area + Lateral surface area
(b) Calculate the total area of the material used by substituting the values of s and h into the equations derived above.
Note: To find the cost of the material used, we need the cost per square meter for each part, which is not given in the question.
To determine the dimensions of the tent that minimize the cost of material used, we need to first express the cost in terms of the dimensions of the tent.
Let's denote:
- x as the length of the equilateral triangle's side (base length)
- h as the height of the right prism
- A as the area of the triangular ends of the tent
(a) The volume of the tent is given as 2.2 m^3, which can be expressed as the product of the base area (A_base) and the height (h) of the prism:
Volume = Base Area * Height
2.2 = A_base * h
We know that the cross-section is an equilateral triangle, so the base area (A_base) can be calculated as follows:
A_base = (sqrt(3) / 4) * x^2
The cost of the material used for the ends and the top of the tent (C_ends_top) can be defined as 1.4 times the cost of the material used for the floor (C_floor):
C_ends_top = 1.4 * C_floor
To express the cost in terms of the dimensions of the tent, we need to determine the total area of material for each component of the tent:
- For the triangular ends: 2 * A_base
- For the top: A_base
The total cost (C_total) can then be calculated as:
C_total = C_floor * A_floor + C_ends_top * (2 * A_base + A_base)
= C_floor * A_floor + C_ends_top * 3 * A_base
Given that the volume is fixed at 2.2 m^3, we need to express the cost in terms of one variable. Let's use x, the side length of the equilateral triangle:
A_base = (sqrt(3) / 4) * x^2
h = 2.2 / A_base
Substituting these expressions into the total cost equation:
C_total = C_floor * A_floor + C_ends_top * 3 * A_base
= C_floor * x^2 + C_ends_top * 3 * (sqrt(3) / 4) * x^2
Now we have the cost equation in terms of x alone. To find the dimensions that minimize the cost, we can differentiate with respect to x, set it equal to zero, and solve for x:
dC_total/dx = 2C_floor * x + 3C_ends_top * (sqrt(3) / 2) * x = 0
Simplifying the equation:
C_floor * x = C_ends_top * (sqrt(3) / 2) * x
Since x ≠ 0, we can divide both sides by x:
C_floor = C_ends_top * sqrt(3) / 2
(b) The total area of material used is the sum of the areas of the floor, the two triangular ends, and the top:
Total Area = A_floor + 2 * A_base + A_base
= A_floor + 3 * A_base
To calculate the total area, we need to express A_floor and A_base in terms of x:
A_floor = x^2
A_base = (sqrt(3) / 4) * x^2
Substituting these values into the total area equation:
Total Area = x^2 + 3 * (sqrt(3) / 4) * x^2
= x^2 + (3sqrt(3) / 4) * x^2
= (4 + 3sqrt(3)) / 4 * x^2
Now we have the total area expression in terms of x. To find the total area, we can substitute the value of x that minimizes the cost into this equation.
(Note: To find the exact dimensions that minimize the cost and calculate the total area, we would need the specific values of C_floor and C_ends_top in monetary units.)