(10 pts) Interpreting Word Problems A cylindrical can is made to hold 1000 cm3 of liquid.
(a) Express the surface area of the can as a function of one variable, making sure to state its domain.2
(b) Suppose that you want to build a can like the one from the previous step, but you want to put three layers
of material on the bottom, two layers in the side, and just one layer on the top. Express
v = πr^2h
so h = 1000/(πr^2)
(a) A = 2πrh + 2πr^2 = 2πr^2 + 2πr(1000/(πr^2)) = 2πr^2 + 2000/r
(b) A = 3*πr^2 + 2*2000/r + 1*πr^2 = 4πr^2 + 4000/r
(a) Let's denote the radius of the cylindrical can as "r" and the height as "h".
The formula for the surface area of a cylindrical can is given by:
Surface Area = 2πr² + 2πrh
Given that the can is made to hold 1000 cm³ of liquid, we can also write:
πr²h = 1000
The surface area can be expressed as a function of "r" or "h", but let's express it in terms of "r" and "h" so that we have a function of one variable.
From the equation πr²h = 1000, we can rearrange it to solve for "h":
h = 1000 / (πr²)
Substituting this value of "h" into the surface area formula, we get:
Surface Area(r) = 2πr² + 2πr(1000 / πr²)
= 2πr² + 2000 / r
The domain of this function depends on the dimensions of the can. In general, the radius "r" and height "h" should be positive values since we cannot have negative dimensions for a can. However, there may be practical limitations on the maximum and minimum values of "r" and "h" based on the manufacturing process or other constraints. Without specific constraints, the domain of the function can be assumed to be all positive real numbers.
(b) To express the surface area of the can with the specified layers, we need to modify the surface area formula according to the given layering pattern.
For the bottom:
To have three layers of material on the bottom, the original radius "r" needs to be increased by three times, resulting in a new radius of "4r".
For the sides:
To have two layers on the sides, the height remains the same, so we keep "h".
For the top:
To have just one layer on the top, the original radius "r" is used.
Using these modifications, the surface area formula becomes:
Surface Area(r) = 2π(4r)² + 2π(4r)(h) + πr²
Simplifying:
Surface Area(r) = 32πr² + 8πrh + πr²
The domain for this modified surface area function remains the same as in part (a) since we are using the same dimensions and just modifying the layering pattern.
To solve this problem, we need to understand the formulas for surface area and volume of a cylinder:
1. Surface area of a cylinder:
The surface area of a cylinder consists of three parts: the top and bottom circles, and the curved surface area (or lateral area). The formulas for these are:
- Top/bottom circles: A_circle = πr^2 (where r is the radius of the cylinder)
- Curved surface area: A_curved = 2πrh (where r is the cylinder's radius and h is its height)
2. Volume of a cylinder:
The volume of a cylinder is given by the formula:
V = πr^2h (where r is the radius of the cylinder and h is its height)
Now let's answer the given questions step by step:
(a) Express the surface area of the can as a function of one variable, making sure to state its domain:
Since the surface area consists of the top and bottom circles and the curved surface area, we can express the function as:
A(r, h) = 2πr^2 + 2πrh (where r is the radius and h is the height of the cylinder)
To state the domain, we need to consider the restrictions of the problem. In this case:
- The radius (r) must be positive, as it represents a physical dimension.
- The height (h) must also be positive, as it represents the height of the cylinder.
Therefore, the domain for this function is r > 0 and h > 0.
(b) Suppose that you want to build a can like the one from the previous step, but you want to put three layers of material on the bottom, two layers on the side, and just one layer on the top:
To express the new surface area function, we need to multiply the areas of the respective sections by the number of layers.
For the bottom:
Since we want three layers, we multiply the area of one circle by 3:
A_bottom = 3(πr^2)
For the top:
Since we want only one layer (no layers added to the top), we multiply the area of one circle by 1:
A_top = 1(πr^2)
For the side:
Since we want two layers, we multiply the curved surface area by 2:
A_side = 2(2πrh) = 4πrh
The total surface area (A_total) would be the sum of the areas of the bottom, top, and side layers:
A_total = A_bottom + A_top + A_side
= 3(πr^2) + 1(πr^2) + 4πrh
Remember that the domain for this function is the same as in part (a): r > 0 and h > 0.