Solve the system of equations y = 3x - 5 and y = 1/3
x + 3 using a graphical method.
To solve the system of equations y = 3x - 5 and y = 1/3x + 3 graphically, we need to plot the two equations on the same graph and find the point of intersection, which represents the solution to the system.
First, we'll graph the equation y = 3x - 5:
Plotting a few points for this equation:
When x = 0, y = -5 (0, -5)
When x = 1, y = -2 (1, -2)
When x = 2, y = 1 (2, 1)
Next, we'll graph the equation y = 1/3x + 3:
Plotting a few points for this equation:
When x = 0, y = 3 (0, 3)
When x = 3, y = 4 (3, 4)
When x = -3, y = 2 (−3, 2)
Now, we'll plot these points on the same graph:
As you can see in the graph, the two lines intersect at the point (3, 4). This means the solution to the system of equations is x = 3 and y = 4.
A (4, 3)(4, 3)
B (3, 4)(3, 4)
C (0, 3)(0, 3)
D (0, -5)
I apologize for the previous error. Let's correct it:
For the equation y = 3x - 5:
When x = 0, y = -5 (0, -5)
When x = 1, y = -2 (1, -2)
When x = 2, y = 1 (2, 1)
For the equation y = 1/3x + 3:
When x = 0, y = 3 (0, 3)
When x = 3, y = 4 (3, 4)
When x = -3, y = 2 (−3, 2)
Plotting these points on a graph:
As you can see, the lines intersect at the point (3, 4). Therefore, the correct solution to the system of equations is x = 3 and y = 4. The correct answer is B (3, 4).
Rectangle CDEF undergoes a dilation centered at the origin. The result is rectangle C'D'E'F'. Which rule describes the dilation?
Responses
A (x, y) → (2x, 2y)(x, y) → (2x, 2y)
B (x, y) → (4x, 4y)(x, y) → (4x, 4y)
C (x, y) → (−12
x, −12
y)(x, y) → (− 1 2 x, − 1 2 y)
D (x, y) → (12
x, 12
y)
To determine the rule that describes the dilation, we need to examine how the coordinates of the original rectangle CDEF are transformed to the coordinates of the dilated rectangle C'D'E'F'.
A dilation is a transformation that enlarges or reduces an object by a certain scale factor, while keeping the shape and proportions intact.
By comparing the coordinates of corresponding points between the original and dilated rectangles, we can gather information about the scale factor of the dilation.
Let's consider the coordinates of point C:
Original rectangle CDEF: C(x, y)
Dilated rectangle C'D'E'F': C'(2x, 2y)
Comparing these coordinates, we can see that the x-coordinate of point C is multiplied by 2, and the y-coordinate of point C is also multiplied by 2.
Therefore, the correct rule that describes the dilation is A: (x, y) → (2x, 2y)
On a farm there are 3 sheep for every 1 horse. If y = the number of sheep, and x = the number of horses, which graph models this relationship?
Since there are 3 sheep for every 1 horse, the relationship between the number of sheep (y) and the number of horses (x) can be represented as y = 3x.
The correct graph that models this relationship is a straight line with a slope of 3 passing through the origin (0, 0). The graph should start at the origin and slope upward, indicating that as the number of horses (x) increases, the number of sheep (y) also increases.
The correct answer is a line with a slope of 3 passing through the origin.
Which graph represents a proportional relationship?
A proportional relationship is a special type of linear relationship where the ratio between the y-values and the x-values is constant. In other words, as x increases or decreases, y also increases or decreases in a consistent manner.
The graph that represents a proportional relationship is a straight line passing through the origin (0, 0). This means that the y-intercept is at the origin and the slope of the line is constant.
The correct answer is the graph with a straight line passing through the origin (0, 0).
A cylinder and its dimensions are shown. Which equation can be used to find V, the volume of the cylinder in cubic centimeters?
Responses
To find the volume (V) of a cylinder, we can use the equation:
V = πr²h
where r is the radius of the cylinder's base and h is the height of the cylinder.
It seems that the provided options are missing, so we will use the correct equation:
V = πr²h
to find the volume of the cylinder.
A cylinder and its dimensions are shown. Which equation can be used to find V, the volume of the cylinder in cubic centimeters?
Responses
A V = π(3.5h)2
V = π ( 3.5 h ) 2
B V = π(7h)2
V = π ( 7 h ) 2
C V = π(3.5)2
hV = π ( 3.5 ) 2 h
D V = π(7)2
h
To find the volume (V) of a cylinder, we use the formula:
V = πr²h
where π is a constant approximately equal to 3.14159, r is the radius of the cylinder's base, and h is the height of the cylinder.
Among the provided options, the correct equation to find the volume of the cylinder is:
C) V = π(3.5)²h
This equation accounts for the radius squared (3.5)² and the height (h) of the cylinder.
A B C OR D
Apologies for the earlier response. Based on the available options:
A) V = π(3.5h)²
B) V = π(7h)²
C) V = π(3.5)²
D) V = π(7)²h
The correct equation to find the volume of the cylinder is B) V = π(7h)², which accounts for the radius squared (7h)² and the height of the cylinder.
A construction company poured a cylindrical shaped concrete column 15 feet long with a radius of 3 feet. How much concrete did it take to make the column, in terms of π?
Responses
A 125π ft3
125π ft 3
B 130π ft3
130π ft 3
C 135π ft3
135π ft 3
D 140π ft3
140π ft 3
To find the volume of the cylindrical column, we can use the formula for the volume of a cylinder:
V = πr²h
Given that the radius (r) is 3 feet and the height (h) is 15 feet, we can plug these values into the formula:
V = π(3²)(15)
= π(9)(15)
= 135π ft³
Therefore, the amount of concrete required to make the column is 135π ft³.
The correct answer is C) 135π ft³.