Which explanation can be used to informally derive the formula for the volume of a cone? A. A cone consists of a circle as a bas
2 answers:
You might be interested in
Part A:
x + y = 9
4x + y = 6
To solve this system using substitution, begin by isolating either x or y in the first equation. I'll isolate x.
x + y = 9
x = 9 - y
Substitute this expression (9 - y) into the second equation for x and solve.
4x + y = 6
4(9 - y) + y = 6
36 - 4y + y = 6
36 - 3y = 6
-3y = -30
y = 10
To find x, substitute 10 for y into either of the original equations.
x + y = 9
x + 10 = 9
x = -1
Finally, check all work by substituting the x- and y-values into each original equation.
x + y = 9 --> -1 + 10 = 9 --> 9 = 9 --> True
4x + y = 6 --> 4(-1) + 10 = 6 --> -4 + 10 = 6 --> 6 = 6 --> True
The answer for Part A is x = -1 and y = 10; (-1, 10).
Part B:
For graphing, it's easier to get the equations into slope-intercept form. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. To get our equations into slope-intercept form, we must simply isolate y.
x + y = 9
y = 9 - x
y = -x + 9
4x + y = 6
y = 6 - 4x
y = -4x + 6
Now that we have our slope-intercept equations, we can easily graph them, since their slopes and y-intercepts are readily visible.
Let's start with the y-intercepts. They are (0, 9) and (0, 6). You can plot those points on the graph.
Now, the slopes. The slope of the first line is -1, this means it declines. To plot this, start where you plotted the y-intercept, count one unit down, and then one unit to the right, and plot that point. Continue doing that and connect the dots, and you will have plotted the first line. The slope of the second line is -4, so it also declines. For this line, count four units down, and then one to the right and plot that point. Likewise, continue this and connect the coordinates, and you will have your line. (See attachment for graph.)
The lines do indeed intersect at (-1, 10); our answer is verified by graphing.
1. Remember that the perimeter is the sum of the lengths of the sides of a figure.To solve this, we are going to use the distance formula: 
where
are the coordinates of the first point
are the coordinates of the second point
Length of WZ:
We know form our graph that the coordinates of our first point, W, are (1,0) and the coordinates of the second point, Z, are (4,2). Using the distance formula:
We know that all the sides of a rhombus have the same length, so
Now, we just need to add the four sides to get the perimeter of our rhombus:
We can conclude that the perimeter of our rhombus is
square units.
2. To solve this, we are going to use the arc length formula:
where
is the length of the arc.
is the radius of the circle.
is the central angle in radians
We know form our problem that the length of arc PQ is
inches, so 
, and we can infer from our picture that 
. Lest replace the values in our formula to find the central angle POQ:
Since
, We can conclude that the measure of the central angle POQ is 
3. A cross section is the shape you get when you make a cut thought a 3 dimensional figure. A rectangular cross section is a cross section in the shape of a rectangle. To get a rectangular cross section of a particular 3 dimensional figure, you need to cut in an specific way. For example, a rectangular pyramid cut by a plane parallel to its base, will always give us a rectangular cross section.
We can conclude that the draw of our cross section is:
where
Length of WZ:
We know form our graph that the coordinates of our first point, W, are (1,0) and the coordinates of the second point, Z, are (4,2). Using the distance formula:
We know that all the sides of a rhombus have the same length, so
Now, we just need to add the four sides to get the perimeter of our rhombus:
We can conclude that the perimeter of our rhombus is
2. To solve this, we are going to use the arc length formula:
where
We know form our problem that the length of arc PQ is
Since
3. A cross section is the shape you get when you make a cut thought a 3 dimensional figure. A rectangular cross section is a cross section in the shape of a rectangle. To get a rectangular cross section of a particular 3 dimensional figure, you need to cut in an specific way. For example, a rectangular pyramid cut by a plane parallel to its base, will always give us a rectangular cross section.
We can conclude that the draw of our cross section is: