Modern metal cans, like the one shown, are usually made from steel, and

then covered with a thin layer of tin.
a. Write an inequality representing the amount of sheet metal that can be
used to make a can that is 10 cm tall
b. Graph the inequality
Explain how the graph can be used to determine the minimum amount of metal required for cans of various radii.

I only need help on a and b. I should be fine for B

if it has radius r, then its surface is 2πr(r+10) cm^2

a. To represent the amount of sheet metal that can be used to make a can that is 10 cm tall, we need to consider the circumference of the can. Since the can is made from steel and covered with a thin layer of tin, we assume that the sheet metal used to make the can is only used to form the sides of the can (without considering the base and top).

Let's assume the radius of the can is "r" cm. The circumference of the can is given by the formula: C = 2πr.

Since the sheet metal wraps around the can and forms its sides, the length of sheet metal used will be equal to the circumference of the can. Therefore, the inequality representing the amount of sheet metal that can be used is:

2πr ≤ L

where "r" is the radius of the can in cm, and "L" is the length of sheet metal available.

b. To graph the inequality, we need to plot a graph of the circumference function 2πr and then shade the region below or on the line, depending on the inequality sign.

1. Draw a Cartesian coordinate system on a piece of graph paper.
2. Label the horizontal axis as "r" to represent the radius and the vertical axis as "L" to represent the length of sheet metal used.
3. On the horizontal axis, mark several values for "r" (radius) that you want to consider. For example, let's mark r = 2, 4, 6, 8 cm.
4. Calculate the corresponding values of "L" (length of sheet metal) for each "r" using the equation: 2πr.
5. Plot the points on the graph by pairing the values of "r" and "L."
6. Connect the points with a smooth curve to represent the circumference function 2πr.
7. Shade the region below or on the line (depending on the inequality sign) to represent the area where the amount of sheet metal used is less than or equal to "L."
8. Label the shaded area as the feasible region.

By analyzing the graph, you can determine the minimum amount of metal required for cans of various radii by locating the point where the circumference function 2πr intersects the vertical line representing the length of sheet metal available. This intersection point will give you the minimum radius required for the given length of sheet metal.

To write an inequality representing the amount of sheet metal that can be used to make a can that is 10 cm tall, we need to consider the surface area of the can. The surface area of a cylinder, which is the shape of a can, is given by the formula:

Surface Area = 2πrh + πr^2

Here, h represents the height of the cylinder (10 cm in this case) and r represents the radius of the cylinder. Since the can is made from a sheet of steel covered with a thin layer of tin, we want to minimize the amount of sheet metal used. Therefore, we can write the inequality as:

Surface Area ≤ πr^2 + 2πrh

Now let's graph the inequality:

To graph the inequality, we need to determine which side of the equation represents the region of interest. In this case, since we want to minimize the amount of sheet metal used, we will shade the region below the equation.

First, let's consider the radius (r) values. A radius cannot be negative, so r ≥ 0.

Now, let's consider the height (h) value, which is fixed at 10 cm.

We can choose some arbitrary values for r and calculate the corresponding values of the surface area. For example, let's choose r = 0, 1, 2, 3, and so on. Plug each value of r into the equation of the surface area and calculate the corresponding value of y (surface area).

After calculating multiple points, plot them on a graph. Connect the points with a smooth line, keeping in mind that r ≥ 0. Shade the region below the line to represent the values of r that satisfy the inequality.

Now, you can use the graph to determine the minimum amount of metal required for cans of various radii. By locating the graph on the y-axis, you can find the corresponding y-values (surface areas). The y-value where the line starts represents the minimum amount of metal required for a can with a radius of 0 (which is the smallest possible radius). The graph can then be used to determine the minimum amount of metal required for cans of larger radii by reading the y-values for the desired radii on the x-axis.