MATHEMATICS HIGH SCHOOL

the following table shows the revenue for a company generates based on the increases in the price of the product. What is the y-value of the Vertex of the parabola that models the date?

Answers

Answer 1

Answer:

Answer:

The y-value of the Vertex of the parabola that models the data is 1125.

Step-by-step explanation:

Let the function of parabola is

$f(x)=ax^2+bx+c$

From the given that it is noticed that the parabolic function passing through the points (1,1045), (3,1105) and (5,1125). It means the function must be satisfied by these points.

$1045=a(1)^2+b(1)+c$

$1045=a+b+c$ ....(1)

$1105=a(3)^2+b(3)+c$

$1105=9a+3b+c$ ....(2)

$1125=a(5)^2+b(5)+c$

$1125=25a+5b+c$ ....(3)

On solving (1), (2) and (3) we get,

$a=-5$

$b=50$

$c=1000$

Therefore the equation of parabola is

$f(x)=-5x^2+50x+1000$

The vertex of the parabola is

$((-b)/(2a),f((-b)/(2a)))$

$(-b)/(2a)=-(50)/(2(-5))=5$

$f(5)=1125$

Therefore the vertex is (5,1125) and y-value of the Vertex of the parabola that models the data is 1125.

Answer 2

Answer:

The vertexes of the parabola are, (5, 1125).

Explanation

The table given to us in the problem are the data points that will lie on the parabola, therefore,

Point 1 = (1, 1045)

Point 2 = (3, 1105)

Point 3 = (5, 1125)

Point 4 = (3, 1105)

Point 5 = (1, 1045)

Equation of a Parabola,

We know that the equation of a parabola is given as,

$y = ax^2 +bx+c$

For point 1,

Point 1 = (1, 1045)

Substituting the value in the equation of a parabola,

$1045 = a(1)^2 +(1)b+c\n\n1045 = a+b+c$ ..... equation 1,

For point 2,

Point 2 = (3, 1105)

Substituting the value in the equation of a parabola,

$1105 = a(3)^2 +(3)b+c\n\n1105= 9a+3b+c$ ..... equation 2,

For point 3,

Point 3 = (5, 1125)

Substituting the value in the equation of a parabola,

$1125= a(5)^2 +(5)b+c\n\n1125= 25a+5b+c$ ..... equation 3,

Solving the three equations we get,

a = -5,

b = 50,

c = 1000

Substitute the values in the equation of a parabola,

$y=f(x) = -5x^2 +50x +1000$

How to find Vertexes of a parabola?

To find the vertex of a parabolic equation we bring the equation into the form,

$y = a(x-h)+k\n$ , where h and k are the vertexes of the parabola.

Vertexes of the parabola

Vertex of the Parabola,

$y=f(x) = -5x^2 +50x +1000\n\ny = -5x^2 +50x +1000\n\ny =-5(x^2 -10x)+1000\n\ny =-5(x^2 -10x+25-25)+1000\n\ny =-5(x^2 -10x+25)+ (-5* -25)+1000\n\ny =-5(x^2 -10x +25)+125+1000\n\ny =-5(x^2 -5)^2+1125$

Comparing it to the equation, $y = a(x-h)+k\n$ ,

the vertexes of the parabola are,

(5, 1125)

Learn more about the Equation of a Parabola:

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Answers

To determine which statement is true, first find the measure of the third interior angle. You can do that by subtracting the two known interior angle measures from 180, since interior angle measures of all triangles add to equal 180 degrees.

180 - 65 - 57 = 58

The measure of the third interior angle is 58°.

Now, the largest side of triangle is opposite its largest interior angle; that means the largest side of this triangle is the side opposite the angle that measures 65° - the side representing swim distance. This is true for the middle and smallest sides as well - they're opposite the middle and smallest interior angles. The following shows the sides from largest to smallest as they correspond to angle measures from largest to smallest.

65° = swim
58° = bike
57° = run

Now to finally determine which statement is true.
Statement A is false because run distance is the smallest distance of all three distances.
Statement B is false because, as we've said, run distance is the smallest distance.
Statement C is false because swim distance is greater than bike distance.

That means that the correct answer is D - the bike distance is greater than the run distance.

Ms. Scott wrote a test. Part A had true/false questions, each worth 6 points. Part B had multiple choice questions, each worth 4 points. She made the number ofpoints for Part A equal the number of points for Part B. It was the least number of points for which this was possible,
Answer the following questions.
How many points was each part worth?
points
How many questions did Part A have?
questions
How many questions did Part B have?
questions

Answers

Answer:

1. How many points was each part worth?

- 12 points

2. How many questions did part A have?

- 2 questions

3. How many questions did Part B have?

- 3 questions

Step-by-step explanation:

We can set up our equation like this:

6x = 4y

In the above equation, x is representing the number of true/false questions and y is representing the nymber of multiple choice questions.

Now, the problem tells us that they want the least number of points possible so we know we need to use low numbers.

Since 6 is higher than 4, it's easier to go off of there.

6 x 1 = 6 4 is too big to go into 6 so we will move on.

6 x 2 = 12 4 goes into 12 3 times so we can use this.

Now that we've figured this out, we can put it in our equation:

6(2) = 4(3)

In the above equation, we can see that I've put 2 in for x because we multiplied 6 by 2 to get 12. I also put 3 in for y because we multiplied 4 by 3.

Now we can start with the questions:

1. How many points was each part worth?

Each part was worth 12 points because we can multiply 6 by 2 and get 12 or 4 by 3 and get the same thing

2. How many questions did part A have?

Part A had 2 questions because this is what x was when we multiplied by 6

3. How many questions did Part B have?

Part B had 3 questions because this is what y was when we multiplied by 4

Hope this helps!!

Final answer:

Each part is worth 12 points. Part A has 2 questions. Part B has 3 questions.

Explanation:

The problem states that the number of points for Part A is equal to the number of points for Part B, and we need to find the least number of points for which this is possible. Let's represent the number of questions in Part A as x. Since each true/false question is worth 6 points, the total points for Part A will be 6x. Similarly, let's represent the number of questions in Part B as y. Since each multiple choice question is worth 4 points, the total points for Part B will be 4y. To find the least number of points for which the two parts are equal, we need to find the smallest common multiple of 6 and 4.

The prime factorization of 6 is 2 x 3.

The prime factorization of 4 is 2 x 2.

From the prime factorization, we can see that the least common multiple (LCM) of 6 and 4 is 2 x 2 x 3 = 12.

Therefore, each part is worth 12 points.

To find the number of questions in Part A and Part B, we can substitute 12 for the total points in each part and solve for x and y:

6x = 12

x = 2

4y = 12

y = 3

Learn more about Least Common Multiple here:

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. Let A = (−2, 4) and B = (7, 6). Find the point P on the line y = 2 that makes the total distance AP + BP as small as possible.

Answers

Answer:

P(1,2)

Step-by-step explanation:

There are 2 points.

A(-2,4) and B(7,6)

the point P on the y=2 can also represented as P(x,2)

We can use the distance formula to find the distances AP and BP

$\text{dist} = √((x_1 - x_2)^2 + (y_1 - y_2)^2)$

for AP: A(-2,4) and P(x,2)

$AP = √((-2 - x)^2 + (4 - 2)^2)$

$AP = √((-2 - x)^2 + 4)$

$AP = √((-1)^2(2 + x)^2 + 4)$

$AP = √((2 + x)^2 + 4)$

for BP: B(7,6) and P(x,2)

$BP = √((7 - x)^2 + (6 - 2)^2)$

$BP = √((7 - x)^2 + 16)$

the total distance AP + BP will be

$√((2 + x)^2 + 4)+√((7 - x)^2 + 16)$ (plot is given below)

Our task is to find the value of x such that the above expression is small as possible. (we can find this either through plotting or differentiating)

If you plot the above equation, the minimum point of the curve will be clearly visible, and it will be at x = 1. Hence, the point P(1,2) is such that the total distance AP + BP is as small as possible.

Final answer:

The point P that makes the total distance AP + BP smallest on the line y=2 is given by the x-coordinate of the midpoint of A and B because the shortest distance is in a straight line. Therefore, the point P is (2.5, 2).

Explanation:

To find the point P on the line y = 2 that makes the total distance AP + BP the smallest, you need to recall that the shortest distance between two points is a straight line. So, ideally, we want to find a point P (x,2) that is on the same vertical line (or x-coordinate) that intersects the line AB at the midpoint.

Step 1: Find the midpoint of A and B. The midpoint M is obtained by averaging the x and y coordinates of A and B: M = ((-2+7)/2 , (4+6)/2) = (2.5, 5).

Step 2: Since line y = 2 is horizontal, the x-coordinate of our point P will stay the same with the midpoint x-coordinate. Therefore, P has coordinates (2.5, 2).

So, the point on the line y = 2 that makes the total distance AP + BP as small as possible is P (2.5, 2).

Learn more about Point here:

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If 28% of a sum is $100.80, what is the sum? A. $360.00
B. $282.24
C. $277.78
D. $129.02

Answers

28% of x = 100.80
.28 × x = 100.80
.28x = 100.80
x = 100.80/.28
x = 360

Answer A

Which statement about the angles in the diagram must be true? A. ∠5 and ∠8 are congruent.
B.∠2 and ∠8 are alternate interior angles. C. ∠4 and ∠7 are corresponding angles.
D. ∠1 and ∠5 are alternate exterior angles.

Answers

Answer:

Step-by-step explanation:

they r located at the same spot & side of the intersecting line on both parallel lines

C

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The means and mean absolute deviations of the individual times of members on two 4 times 400 meter relay track teams are shown in the table below. Means and Mean Absolute Deviations of Individual Times of Members of 4 times 400 meter Relay Track Teams Team A Team B Mean 59.32 s 59.1 s Mean Absolute Deviation 1.5 s 2.4 s What percent of Team B’s mean absolute deviation is the difference in the means? 9% 15% 25% 65%