You are designing a rain gutter made from a piece of sheet metal
that is 1 foot by 5 feet. The gutter is formed by turning up two sides. You
want the rain gutter to have the greatest volume possible. Below is the
polynomial function that represents the volume of the gutter. Use the
graph program by typing in the polynomial function just as it is
to look at the graph. Change your axes to the following:
x max = 4
x min = 4
y max = 1
y min = -4
From the volume formula (V=L*W*H)
V = (5)(x)(1-2x) Where x is representing the amount that should be turned up to create the gutter.
Looking at the graph, which x-value do you think would provide the maximum volume? Why?
a. x = 0
b. x = 2
c. x = ¼
d. x = ½
Looking at the graph, what do you think is the maximum volume? Why?
a. V = 0
b. V = 2
c. V = -1
d. V = ½
v = 0 at x = 0 and 1/2
so, the vertex is at x=1/4
v(1/4) = 5(1/4)(1/2) = 5/8
Thank you very much. That is what I thought it was, but I wasn't sure.
Beaufort Scale and Wind Effects
0 Smoke rises vertically
1 Smoke shows wind direction
2 Wind felt on face
3 Leaves move, flags extend
4 Paper, small branches move
5 Small trees sway, flags beat
6 Large branches sway, flags beat
7 Large branches sway, walking is difficult
8 Twigs break, walking is hindered
9 Slight roof damage
10 Severe damage, trees uprooted
11 Widespread damage
12 Devastation
7.
The Beaufort scale was devised by Frances Beaufort in 1805 to measure wind speeds. The scale is numbered from 0 to 12, and represents wind speeds in the open, 33 feet above ground. The Beaufort scale, B, can be modeled by the function
b=1.9�ã(x+8-5.4)
where x is the speed of the wind in miles per hour.
Sketch the graph of this function.
To find the x-value that would provide the maximum volume, we need to analyze the graph. Here's how:
1. Plot the graph of the polynomial function V = 5x(1-2x) on a graphing program with the specified axis settings. Make sure to set the x-axis limits from 4 to 4 and the y-axis limits from -4 to 1.
2. Look at the shape of the graph and observe the point where the graph reaches its highest point. That is the x-value we are looking for.
3. From the provided answer choices, compare the x-values with the shape of the graph to determine which one corresponds to the highest point.
In this case, the x-value that would provide the maximum volume appears to be:
b. x = 2
By observing the graph, we notice that it reaches its maximum point at x = 2, where the curve is at its peak. This means that turning up 2 feet of the sheet metal is the optimal amount to create the rain gutter with the greatest volume.
Now, let's determine the maximum volume:
1. Substitute the value of x = 2 into the volume formula V = 5x(1-2x).
V = 5(2)(1-2(2))
V = 5(2)(1-4)
V = 5(2)(-3)
V = -30
From the calculation, we find that the maximum volume is:
c. V = -30
The volume of the rain gutter is negative because it represents the area below the x-axis, indicating that the gutter is turned downwards. So the maximum volume of the rain gutter is -30 cubic feet.