A cylindrical tank is to have a capacity of 1000 m³. It is to fit into a foundry that is 12 m wide with a height of 11 m.
The base of the tank will cost half as much as the top. The metal for the side of the tank will cost four fifths as much as the top.
An equation that can be used to model the cost of the tank is C(r)=(3/2)pi r^2+(1600/r) . Determine any restrictions on r. You need not solve further.
To determine any restrictions on r for the cost function, let's analyze the given information systematically.
First, we can identify that the volume of the cylindrical tank is 1000 m³. The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. In this case, we are given the height as 11 m, so we can rearrange the equation to solve for the radius:
1000 = πr² * 11
r² = 1000 / (11π)
r² ≈ 28.5 / π
r ≈ √(28.5 / π)
r ≈ 3.01 m (approximately)
From this calculation, we find that the radius of the tank is approximately 3.01 m.
Next, we are given the equation C(r) = (3/2)πr² + (1600/r) to model the cost of the tank.
Restrictions on r can arise from two aspects:
1. Practicality: The radius of the tank should be feasible to fit within the given dimensions of the foundry, which is 12 m wide. Therefore, the radius of the tank cannot exceed half of the width, i.e., r ≤ 6 m.
2. Mathematical restrictions: The equation can give valid results for any positive value of r. However, the cost function might have some behaviors such as asymptotes or undefined values at certain points. To analyze these behaviors, we would need to further evaluate the equation and analyze its properties.
In summary, the main restriction on r is that it should be less than or equal to half the width of the foundry (r ≤ 6 m) for practicality reasons. Further analysis of the equation C(r) is required to determine any additional restrictions based on mathematical properties.