Answer:
Step-by-step explanation:
To calculate the total cost of the tube's material as a function of the radius (r), we need to determine the dimensions of the tube and then find the cost of each component separately.
Let's assume the height of the cylindrical tube is "h" and the radius of the circular base is "r."
The volume of the cylinder is given as 50 cm³, so we can express this as:
Volume (V) = π * r^2 * h
We need to express the height (h) in terms of the radius (r) and volume (V):
h = V / (π * r^2)
The total cost (C) of the tube's material will be the sum of the cost of the top and bottom (using the circular base area) and the cost of the wall (using the lateral surface area):
Cost (C) = Cost of top and bottom + Cost of wall
Cost of top and bottom = 2 * (Cost per cm² * Area of top and bottom)
Cost of wall = Cost per cm² * Area of lateral surface
Now, let's calculate the areas:
Area of top and bottom = 2 * π * r^2
Area of lateral surface = 2 * π * r * h
Substitute the expression for "h" from step 2 into the area of the lateral surface:
Area of lateral surface = 2 * π * r * (V / (π * r^2))
Area of lateral surface = 2 * (V / r)
Now, let's put everything together:
Cost of top and bottom = 2 * (0.15/cm² * 2 * π * r^2)
Cost of top and bottom = 0.3 * π * r^2
Cost of wall = 0.10/cm² * 2 * (V / r)
Cost of wall = 0.20 * (V / r)
Total Cost (C) = 0.3 * π * r^2 + 0.20 * (V / r)
Finally, when r = 2 cm, we can substitute this value into the total cost equation to approximate the cost:
C ≈ 0.3 * π * (2 cm)^2 + 0.20 * (50 cm³ / 2 cm)
C ≈ 1.884 π + 5 cm²
Now, we can approximate the cost of the tube if r = 2 cm:
C ≈ 1.884 π + 5 ≈ 11.889 cm²
Please note that the cost is in terms of square centimeters (cm²). If you need the cost in dollars, you would need to convert using the given cost per cm².