The soft drink vendor's maximum profit per day is $5250, and he must sell 1500 cans of soda pop to reach the maximum profit.
To find the maximum profit per day for the soft drink vendor, we need to use the formula P(x) = 0.001x^2 + 3x + 1800, where x is the number of cans of soda pop sold in one day.
To find the maximum profit, we need to find the vertex of the parabola represented by the profit function. The x-coordinate of the vertex is given by -b/2a, where a = 0.001 and b = 3. Plugging in these values, we get x = -3/(2*0.001) = -1500.
Since the soft drink vendor cannot sell a negative number of cans, we know that the maximum profit occurs at the closest whole number to x = -1500, which is x = 1500.
To find the maximum profit per day, we can plug in x = 1500 into the profit function:
P(1500) = 0.001(1500)^2 + 3(1500) + 1800 = $5250
Therefore, the soft drink vendor's maximum profit per day is $5250, and he must sell 1500 cans of soda pop to reach the maximum profit.
To find the maximum profit per day and the number of cans needed to reach that profit, we'll first need to find the critical point of the given quadratic profit function, P(x) = -0.001x^2 + 3x - 1800.
Step 1: Find the derivative of P(x) with respect to x. This will give us the rate of change of profit as the number of cans sold changes.
P'(x) = -0.002x + 3
Step 2: Set the derivative equal to zero and solve for x. This will give us the critical point where the maximum profit occurs.
-0.002x + 3 = 0
x = 1500 cans
Step 3: Substitute the critical point (x = 1500) back into the profit function P(x) to find the maximum profit.
P(1500) = -0.001(1500)^2 + 3(1500) - 1800
P(1500) = $600
So, the maximum profit per day is $600, and the soft drink vendor must sell 1500 cans to reach the maximum profit.
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The maximum profit per day is $4800, and the vendor must sell 1500 cans of soda pop to reach this maximum profit.
The profit function for the soft drink vendor is given by:
P(x) = [tex]0.001x^2 + 3x + 1800[/tex]
To find the maximum profit, we need to find the vertex of the parabola represented by this function. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
where a = 0.001 and b = 3. Substituting these values, we get:
x = -3 / (2 * 0.001) = -1500
Since the value of x cannot be negative in this context, we know that the maximum profit occurs at x = 1500. To find the maximum profit, we substitute this value of x into the profit function:
P(1500) =[tex]0.001(1500)^2 + 3(1500) + 1800 = $4800[/tex]
Therefore, the maximum profit per day is $4800, and the vendor must sell 1500 cans of soda pop to reach this maximum profit.
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HELPPPPPPPPPPPP PLSSSSSSS
Two lines intersect such that the measure of one of the angles formed by the lines
is five times the measure of another of the angles formed by the lines. What are
the measures of the angles formed by the lines?
The measure of the angles formed by the lines are 30°, 30°, 150° and 150° by definition of vertically opposite angles.
When two straight lines intersect each other at some angle, say line AB intersect line CD, then the opposite angles formed by the two lines are called vertically opposite angles. A pair of vertically opposite angles are equal to each other.
There are two pairs of opposite angles formed by intersection of two straight lines.
The total sum of all the angles formed by intersection of two straight lines is 360°.
Let angle X = x° and angle Y= y° be a pair of opposite angles.
Let the other pair of opposite angles be of angle M = m° and angle N = n°.
Thus, by definition of opposite angles and sum of all angles on intersection of two lines we get,
angle X + angle Y + angle M + angle N = 360°
⇒ x° + y° + m° + n° = 360°
⇒ x° + x° + m° + m° = 360°
⇒ 2( x° + m°) = 360°
⇒ x° + m° = 180° ___ (1)
Also, the measure of one of the angles is five times the measure of another of the angles formed by the lines.
This relation can be written as,
angle X = 5 times of angle M
⇒ x° = 5m° ___(2)
Since angle X and angle M are not a pair of opposite angles but are supplementary angles as observed in equation (1).
Putting equation (2) in equation (1) we get,
5m° + m° = 180°
⇒ 6m° = 180°
⇒ m° = 30°
Thus, x° = 5(30°) = 150°
Hence the angles are 30°, 30°, 150° and 150°.
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Write linear equation in slope intercept form:1/4x+y=-2/7
The given linear equation in slope intercept form is y = -1/4(x) - 2/7.
What is the slope-intercept form?In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;
y = mx + c
Where:
m represents the slope or rate of change.x and y are the points.c represents the y-intercept or initial value.By making "y" the subject of formula, we have the following:
1/4x + y = -2/7
y = -1/4(x) - 2/7
By comparison, we have the following:
Slope, m = -1/4.
y-intercept, c = -2/7.
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Which inequality represents the values of that ensure triangle ABC exists?
A
2x + 4
B
18
6x
O A. < x < 1/1
OB. -< < 1/1
O C.
1 < x < 5
OD. 2 < x < 6
C
The Inequality which ensure triangle exists is 22 > 4x > 7.
What is inequality ?Inequality is defined as the relation between two quantities with the sign of inequality that is >, <, ≤ , ≥ ."
Theorem used In ΔABC,
AB + BC >AC
AC+ BC >AB
AC + AB > BC
According to the question,
In triangle ABC.
AC = 18units
BC = 6x units,
AB = 2x + 4 units
.
Substitute the value in the inequality to ensure triangle exists we get 4x > 7 and 22 > 4x.
From (1) and (2) inequality of triangle we get 22 > 4x > 7.
Hence, Inequality which ensure triangle exists is 22 > 4x > 7.
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Which inequality represents the values of that ensure triangle exists? a triangle with these side lengths: AC = 18 units, BC = 6x units, and AB = 2x + 4 units
FIND ZC OF THIS CIRCLE
The value of ZC is 16 units for the given circle and its identities.
What is a tangent of a Circle?A straight line that touches the circle at a single point—the point of tangency—is the tangent of a circle. The digression is opposite to the span of the circle at the place of juncture, and it broadens outward from the circle in the two bearings.
In the circle two external secant segments are LE and ZE, according to the external secant theorem;
⇒ LE × DE = ZE × CE (Here, ZE = ZC + 2)
⇒ 9 × 4 = (ZC + 2) × 2
⇒ ZC + 2 = 18
⇒ ZC = 18 - 2
⇒ ZC = 16
Therefore, the value of ZC is 16 units.
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A piece of machinery is designed in a shape of a triangular prism. It must be coated in Teflon before it is placed in the machine. How many square inches of Teflon will be needed to completely to cover the machinery
We would need 168 square inches of Teflon to completely cover the triangular prism machinery.
To calculate the surface area of a triangular prism, we need to find the area of each face and then add them up. Let's assume that the triangular bases of the prism have base b and height h, and that the length of the prism is l.
The surface area of the triangular bases is (1/2)bh for each base, so the total area is (1/2)bh + (1/2)bh = bh.
The lateral surface area of the prism is the area of the three rectangular faces. Each rectangular face has length l and height h, so the total lateral surface area is 3lh.
Therefore, the total surface area of the triangular prism is
bh + 3lh
To determine how much Teflon is needed to cover the machinery, we need to know the dimensions of the triangular prism. Once we have those dimensions, we can calculate the surface area using the formula above and then use that value to determine the amount of Teflon required.
Let's say the base of the triangular prism is 4 inches and the height is 6 inches, and the length of the prism is 8 inches. Then the surface area of the triangular prism is
bh + 3lh
= (4 in. x 6 in.) + 3(8 in. x 6 in.)
= 24 in² + 144 in²
= 168 in²
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Adding and subtracting decimals
1) 34.12 - 41.23 + 14.01
2) -34.12 + 41.23 - 14.01
Combine all positive numbers:
Combine all negative numbers:
Rewrite:
(look at the picture)
Answer:
1Step-by-step explanation:
PLEASE is due very soon please explain 50pts
Answer:
3
Step-by-step explanation:
Because same numbers = same y intercept but I'm not sure
Answer:
1) The image has the same slope, different y-intercept
Step-by-step explanation:
You want the relationship between the line 3x -5y = 4 and its image dilated by a factor of 5/3 centered at the origin.
DilationDilation about the origin multiplies every coordinate by the dilation factor. Dilation does not change slopes or angles, but it changes lengths by the dilation factor.
Dilation of a lineWhen the equation of a line is written in standard form, the dilation of it effectively multiplies the constant by the dilation factor.
The given line has intercepts ...
ax +by = c
x-intercept = c/a = 4/3
y-intercept = c/b = -4/5
If these values are multiplied by the dilation factor 5/3, they become ...
x-intercept' = (5/3)(4/3) = 20/9 = 2 2/9
y-intercept' = (5/3)(-4/5) = -4/3 . . . . . note the y-intercept has changed
SlopeThe slope of the given line can be determined from its intercepts:
m = (-y intercept)/(x intercept) = -(-4/5)/(4/3) = 3/5
Since both of these intercepts are multiplied by the same dilation factor, their ratio remains unchanged. The slope of the dilated line is the same.
EquationEffectively, the dilated line's equation is ...
3x -5y = 4(5/3)
3x -5y = 20/3
__
Additional comment
The original (dashed) line and its dilated image are shown in the attachment.
The coordinates of P are (-3, 8) and the coordinates of Q are (9, -2).
Find the equation of the line parallel to PQ that passes through the point (6, -1).
The equation of the line parallel to PQ that passes through the point (6, -1) is y = (-5/6)x + 4.
What is the equation of the line parallel to PQ that passes through the point (6, -1)?The formula for equation of line is expressed as;
y = mx + b
Where m is slope and b is y-intercept.
Given that: the coordinates of P are (-3, 8) and the coordinates of Q are (9, -2).
We can find the slope of line PQ.
m = (y2 - y1) / (x2 - x1)
Where (x1, y1) and (x2, y2) are the coordinates of points P and Q, respectively.
m = (-2 - 8) / (9 - (-3))
m = -10 / 12
m = -5/6
Since we want the equation of a line parallel to PQ, it will have the same slope.
Hence, the equation of the line can be written as:
y - y1 = m(x - x1)
where (x1, y1) is the point (6, -1) and m is the slope we found earlier.
Plugging in the values, we get:
y - (-1) = (-5/6)(x - 6)
y + 1 = (-5/6)x + 5
y = (-5/6)x + 5 - 1
y = (-5/6)x + 4
Therefore, the equation of the parallel line to PQ is y = (-5/6)x + 4.
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How to calculate a derivative using implicit differentiation
To calculate a derivative using implicit differentiation, follow these steps:
1. Write down the equation:
2. Differentiate both sides with respect to x:
3. Solve for dy/dx or y':
4. Simplify the result:
Write down the equation:
Start by writing down the equation involving both variables x and y, where y is an implicit function of x.
Differentiate both sides with respect to x: Apply the chain rule, product rule, or quotient rule, as needed, while differentiating both sides of the equation with respect to x.
Remember that when differentiating any term involving y, you'll also need to multiply by the derivative of y with respect to x, denoted as dy/dx or y'.
Solve for dy/dx or y': After differentiating both sides, you will have an equation involving dy/dx or y'.
Solve for this term to find the derivative.
Simplify the result: Simplify your answer as much as possible, if needed.
Remember to use implicit differentiation when the equation is not easily solved for y in terms of x.
It's a powerful technique that helps find derivatives in situations where explicit differentiation isn't feasible.
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NEED HELP FAST PLEASE HURRY INDEED OF HELP
Answer:
Step-by-step explanation:
Hopefully this is right.
Area = (pi)r²
Which is pi times the the radius squared
The radius is 2, so 2² is 4, So we multiply 3.14, which is pi, by 4.
Multiply 3.14 by 4, and the answer is 12.56, the question says round to the nearest tenth, so the final answer will be 12.6
My math teacher back in 7th grade gave us this sentence to help us memorize the equations.
Cherry Pie Delicious (C=(pi)d), Apple Pie Are Too (A=(pi)r²)
Do the same with the other two questions, and the answer for question two will be 3.14, rounded to the nearest tenth will be 3.1
Question three do the same as well, and the answer will be 660.185, rounded will be 660.2
Hope this helps!
A rectangular prism has a length of 10 inches, a width of 4 inches, and a height of 3.7 inches. What is the volume of this rectangular prism?
Answer:
I believe it's 148.
Step-by-step explanation:
You must do L x W x H to find the volume of a rectangular prism or cube.
The set of values for x that satisfies a quadratic inequality is x < -0.5 or x > 1.5
Write down a possible quadratic inequality. Can u show me working it how to do this please. Thanks
Answer:
To write a possible quadratic inequality, we first need to determine which quadratic function has roots at x = -0.5 and x = 1.5.
We know that the solution to the quadratic inequality is x < -0.5 or x > 1.5, which means that the parabola associated with our quadratic function should not intersect the x-axis between x = -0.5 and x = 1.5.
One quadratic function with roots at x = -0.5 and x = 1.5 is:
(x + 0.5)(x - 1.5) ≤ 0
To see why this is the case, we can use a graphing calculator to graph the quadratic function y = (x + 0.5)(x - 1.5) and observe that the function is negative (i.e., has a y-value less than zero) when x is less than -0.5 or greater than 1.5.
Alternatively, we can use the fact that the product of two factors is less than or equal to zero if and only if one factor is positive and the other is negative.
Since (x + 0.5) is positive for x > -0.5 and negative for x < -0.5, and (x - 1.5) is positive for x > 1.5 and negative for x < 1.5, the inequality (x + 0.5)(x - 1.5) ≤ 0 is true for x < -0.5 or x > 1.5, but false for -0.5 < x < 1.5.
Therefore, a possible quadratic inequality that satisfies the given set of values for x is:
(x + 0.5)(x - 1.5) ≤ 0
it is known that roughly 2/3 of all human beings have a dominant right foot or eye. is there also right-sided dominance in kissing behavior? an article reported that in a random sample of 127 kissing couples, both people in 79 of the couples tended to lean more to the right than to the left.
It is possible that there is right-sided dominance in kissing behavior, based on the reported study. The fact that in a random sample of 127 kissing couples, both people in 79 of the couples tended to lean more to the right than to the left suggests a pattern of right-sided dominance.
However, it is important to note that this study alone is not conclusive evidence of right-sided dominance in kissing behavior.
Further research would be needed to confirm whether there is a consistent pattern of right-sided dominance in kissing behavior across different populations and cultures. It is also possible that other factors, such as social or psychological factors, could influence the direction in which people lean during a kiss. Therefore, it is important to approach this topic with caution and not draw any firm conclusions based on a single study.
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What happens when the function f(x) = 2 sin(2) is transformed by the rule g (w) = - f(x)?
• f(x) is stretched away from the x-axis by a factor of 2 and is reflected over the x-axis.
O f(x) is reflected over the y-axis.
O f(x) is reflected over the x-axis.
• f(x) is compressed toward the y-axis by a factor of 1/2 and is reflected over the y-axis
Answer:
• f(x) is stretched away from the x-axis by a factor of 2 and is reflected over the x-axis.
Step-by-step explanation:
The function f(x) = 2sin(2x) has an amplitude of 2 and a period of π/2. When we apply the rule g(w) = - f(x), we are reflecting the graph of f(x) over the x-axis and then taking its opposite. This means that the amplitude of g(w) will still be 2, but the graph will be reflected and stretched away from the x-axis by a factor of 2.
I hope it helps you
What is the area of a right triangle with a height of 6-yards and a base of 22 yards?
O 34 yds²
O 68³
yds²
O 132 yds²
O
137/ yds²
Answer:
68 3/4 yds
Step-by-step explanation:
trust me :)
Answer:
see below
Step-by-step explanation:
area of triangle formula = 1/2 base * height
so 1/2*6*22 =66
not sure why there's no choice of 66
Kevin has a remote control car. Every 3/40 of a second, the remote control car travels 3/ 8 of a foot. How many feet per second does the remote control car travel?
The number of feet per second that the remote control car travels is: 5 ft/s
How to divide fractions?The formula to find the speed from distance and time is:
Speed = Distance/time
Now, we are given:
Time: t = 3/40 seconds
Distance = 3/8 ft
Thus:
Speed = (3/8)/(3/40)
Speed = (3/8) * (40/3)
Speed = 5 ft/s
Thus, that is the speed that the remote control car travels
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Draw triangle PQR where PQ has length 4 cm PR has length 3.5 cm and angle RPQ is 32° 
What is the value of x in the equation 2x + 3 = 11?
Answer: x=4
Step-by-step explanation:
Isolate x using opposite operations.
2x + 3 = 11
2x = 8
x = 4
Answer:
4
Step-by-step explanation:
subract 3 from 11
2x=8
divide by 2
x=4
A rectangle and a triangle below have the same height and base measurements. Find the area of the rectangle and the right triangle, and then compare the areas
The area of rectangle is ____ square inches. The area of the triangle is __ square inches . The area of the triangle is ___ the area of the rectangle
The area of the rectangle is 48 in².
The area of the triangle is 24 in².
The area of the triangle is half the area of the rectangle.
How to calculate the area of a rectangle and a triangle?To calculate the area of a rectangle, we multiply the length of the rectangle by its width. And to calculate the area of a triangle we multiply the base of the triangle to the height of the triangle and divide by 2.
To calculate the area of the rectangle, we use the formula below
Formula:
A = LW....................... Equation 1Where:
A = Area of the rectangleL = Length of the rectangleW = Width of the rectangleFrom the question,
Given:
L = 6 inW = 8 inSubstitute into equation 1
A = 6×8A = 48 in²Also, the area of the triangle is given by the formula below
A' = bh/2.................. Equation 2Where:
A' = Area of the triangleb = Base of the triangle = 6 inh = Height of the triangle = 8 inSubstitute into equation 2
A' = (6×8)/2A' = 24 in²Hence, the area of the triangle is half the area of the rectangle.
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Deduce the 3-digit secret number.
[A cow means a number is correct in value but in the wrong
position. A bull indicates that a number is both correct in
value and in the correct position. I. E. 2 cows and 1 bull wou
indicate that all three numbers were correct but two were in
the wrong positions. ]
If "cow" means that number is correct in value but in the wrong position, then the three-digit secret number is 714.
A "cow" means a digit is "correct in value" but in the "wrong position".
A "bull" means a digit is both "correct in value" and in the "correct position".
⇒ For the first guess (751),
There is 1 cow and 1 bull, which means that one digit is "correct in value" and in the "correct position" (1 bull), and another digit is "correct in value" but in the "wrong position" (1 cow).
⇒ For the second-guess (574),
There is 1 cow and 1 bull, which indicates that "one-digit" is "correct in value" and in the "correct position", and "another-digit" is "correct in value" but in the "wrong-position".
⇒ For the third guess (317),
There is 1 cow and 1 bull, which means that "one digit" is "correct in value" and in the "correct position", and "another digit" is "correct in value" but in the "wrong position".
So, Based on these clues, we can say that the correct secret number has 1 cow and 1 bull.
This means that two digits in the secret number are correct in value, with one digit in the correct position (1 bull), and another digit in the wrong position (1 cow). So, the secret number is likely 714, because it satisfies these conditions.
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The given question is incomplete, the complete question is
Deduce the 3-digit secret number.
[A cow means a number is correct in value but in the wrong position. A bull indicates that a number is both correct in value and in the correct position. i.e. 2 cows and 1 bull would indicate that all three numbers were correct but two were in the wrong positions. ]
Guess Secret Number Cows Bulls
1st 751 1 1
2nd 574 1 1
3rd 317 1 1
Find the area of the circle. Round your answer to the nearest hundredth. Use $3.14$ or $\frac{22}{7}$
for $\pi$ .
area: about
ft2
The area of the given circle is 1.76625 square feet.
How to find the area of the circle?Remember that the area of a circle of radius R is given by the formula:
A = pi*R²
Where pi = 3.14
Here we know the diameter (2 times the radius) of the circle, which is 1.5ft, then the radius is:
R = 1.5ft/2 = 0.75ft
Then the area of the circle is:
A = 3.14*(0.75ft)²
A = 1.76625 ft²
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A triangle has two sides measuring 4 cm and 8 cm. Which lengths cannot be the length of the third side
The length of the third side of a triangle cannot be 15 cm
The correct answer is an option (b)
We know that the triangle inequality theorem states that the sum of any two sides of a triangle is greater than or equal to the third side.
For example if a, b, c represents the side of a triangle then by above theorem a + b ≥ c
Here, the two sides of a triangle measure 4 cm and 8 cm.
Let a = 4 cm and b = 8 cm and c represents the third side of a triangle.
Using triangle inequality theorem,
a + b ≥ c
4 + 8 ≥ c
12 ≥ c
This means that the third side of the triangle must be less than or equal to 12 cm.
Thus, the correct answer is an option (b)
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The complete question is:
A triangle has two sides measuring 4 cm and 8 cm. Which lengths cannot be the length of the third side
a) 7 cm
b) 15 cm
c) 10 cm
d) 5 cm
A teacher surveyed her class after they had taken a vocabulary test. Eighteen of the students claimed they had studied
at least one hour for the test. The remaining twelve students admitted that they had not studied for the test at all. The
test results (expressed as a percent) for the two groups are shown below.
Studied: 88, 100, 94, 79, 92, 100, 95, 83, 89, 99, 100, 91, 89, 95, 100, 93, 96, 84
Did Not Study: 82, 72, 45, 91, 58, 83, 65, 87, 90, 77, 73, 89
1. Calculate the range and Interquartile range for each set of data.
2. Using complete sentences, compare the data sets based on your results in part A.
NEED THIS ASAP PLEASE
How many times can 3.6 go into 80
Answer:
22.2
Step-by-step explanation:
Answer:
3.6 can go into 80 22 times.
Step-by-step explanation:
Let an be the number of ways to climb n stairs Identify the recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time. O an = an + 1 + an + 2 O an = an-1+an - 2 O an= an-1+ an-2
The total number of ways to climb n stairs is the sum of the number of ways to climb n-1 stairs and the number of ways to climb n-2 stairs.
What is recurrence relation?An equation that represents a sequence based on a rule is called a recurrence relation. Finding the following term (next term) based on the prior term (previous term) is made easier.
The recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time is:
an = an-1 + an-2
This is because to climb n stairs, the person can either take one stair from n-1 stairs and climb the remaining n-1 stairs, or take two stairs from n-2 stairs and climb the remaining n-2 stairs.
Therefore, the total number of ways to climb n stairs is the sum of the number of ways to climb n-1 stairs and the number of ways to climb n-2 stairs.
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Mr. Johnson took the price of a watch that was listed at $60 and marked it up by 30%. After the mark up, What is the selling price of the watch?
Answer:
$78
Step-by-step explanation:
60+30% = 78
...................
Question 4 of 10
Does this table represent a function? Why or why not?
X
2
3
4
5
5
y
1
4
4
2
5
OA. Yes, because there is the same number of x-values as y-values.
OB. Yes, because every x-value corresponds to exactly one y-value.
C. No, because one x-value corresponds to two different y-values.
OD. No, because two of the y-values are the same.
Does this table represent a function: C. No, because one x-value corresponds to two different y-values.
What is a function?In Mathematics and Geometry, a function can be defined as a mathematical equation which is typically used for defining and representing the relationship that exists between two or more variables such as an ordered pair.
This ultimately implies that, all functions are used in mathematics to uniquely map an input variable (domain or independent values) to an output variable (range or dependent values).
Based on the table shown above, we can reasonably infer and logically deduce that it does not represent a function because the input values (domain or independent values) are not uniquely mapped to the output values (range or dependent values);
3 → 4
4 → 4
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exploratory data analysis 1. when considering the research question, identify the two variables predictor (explanatory) variable: response variable: 2. use minitab to first create a scatterplot. minitab> graph > scatterplot how would you characterize the association between these two variables? a. linear or nonlinear? b. negative, none, or positive?
Identify the predictor and response variables, then create a scatterplot in Minitab. Characterize the association between the variables as linear or nonlinear and negative, none, or positive.
In exploratory data analysis, the first step is to identify the research question and the two variables of interest. The predictor variable (also known as the explanatory variable or independent variable) is the one that is used to explain or predict the outcome of the response variable (also known as the dependent variable).
After identifying the two variables, one can use software such as Minitab to create a scatterplot of the data. The scatterplot helps to visualize the relationship between the two variables.
To characterize the association between the two variables, we would first determine if the relationship appears to be linear or nonlinear. If the relationship is roughly linear, we would then determine if it is negative (meaning that as one variable increases, the other tends to decrease), positive (meaning that as one variable increases, the other tends to increase), or none (meaning that there is no clear trend).
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Which is the only congruence theorem that cannot be use to prove these triangles to be congruent?
A
SAS
B
AAS
C
SSS
D
ASA
The only congruence theorem that cannot be used to prove these triangles to be congruent is option B, AAS (Angle-Angle-Side).
The SAS (Side-Angle-Side), SSS (Side-Side-Side), and ASA (Angle-Side-Angle) congruence theorems all involve matching corresponding sides and angles of two triangles. However, AAS only matches two angles and a non-included side, which is not enough to guarantee congruence in all cases.
In fact, the AAS congruence theorem only works if the given non-included side is between the two given angles, forming a "triangle sandwich" with the two given angles as the bread and the non-included side as the filling.
If the non-included side is not between the two given angles, then it is possible to have two non-congruent triangles with the same angle measures, but different side lengths.
Therefore, the AAS congruence theorem cannot be used to prove all cases of triangle congruence, making it the only option that cannot be used to prove these triangles to be congruent
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