Question 3
The first five triangular numbers are shown, where n represents the number of dots in the base of the figure, and d (n)
represents the total number of dots in the figure.
1 3 6
15
When n = 1, there is 1 dot. When n = 2, there are 3 dots. When n = 3, there are 6 dots. Notice that the total number of dots
d (n) increases by n each time.
Use induction to prove that d (n) = n(n+1)
10
Part A
Prove the statement is true for n = 1.

Answer:

22 and 23

Step-by-step explanation:

The first step is to make an equation that models this problem. Since the two numbers are consecutive, make them [tex]x[/tex] and [tex]x+1[/tex].

Since it says three times the larger number is 47 more than the smaller number, that means

[tex]3 (x+1) = x+47[/tex]

[tex]3x+3 = x+47[/tex]

[tex]2x+3=47[/tex]

[tex]2x=44[/tex]

[tex]x=22[/tex]

The smaller number is 22, and the larger number is 23.

The number of f(x) = g(x) solutions is an option (2) The right answer is 2, thus.

A graph is a diagram or visual representation that organizes the presentation of facts or values. The collection of all points in the plane with the form (x, f(x)) that make up a function f's graph.

For the predicament,

f(x) = In(x + 2) and g(x) = e(x - 1) throughout the range [(-2,3].

In the graph below,

plot the two functions f(x) and g(x).

These two functions cross at two locations in the range [(-2,3]).

The function's solution is revealed at the intersection point.

Consequently, we might say that the number of solutions.

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Answer:

[tex](1, -1)[/tex] and [tex](-\frac{1}{5}, \frac{7}{5})[/tex]

In decimal form this would be (1, -1) and (-0.2, 1.4)

Step-by-step explanation:

Basically here we are trying to solve a system of 2 equations. The first one is a straight line and the second is a circle. This can be done graphically and graph is attached. There will be two points where the line cuts the circle so two solution sets

The equations are

Line Equation [tex]y = 1 - 2x[/tex]    (1) and

Circle Equation [tex]x^2+ y^2 = 2[/tex]   (2)

[tex]y=1-2x[/tex]

Plug this value of y into the second equation for the circle
[tex]x^2 + (1-2x)^2= 2[/tex]

Apply the perfect squares formula [tex]\left(a-b\right)^2=a^2-2ab+b^2[/tex]

[tex](1-2x)^2 = 1^2-2\cdot \:1\cdot \:2x+\left(2x\right)^2\\= 1-4x+4x^2\\\\[/tex]

So equation (2) becomes
[tex]1-4x+4x^2 +x^2 = 2 \\\\\\5x^2 -4x - 1 = 0[/tex]

Solve using the quadratic formula
[tex]x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]

Here a is the coefficient of x ², b is the coefficient of x and  c is the constant

So
[tex]x_{1,\:2}=\frac{-\left(-4\right)\pm \sqrt{\left(-4\right)^2-4\cdot \:5\left(-1\right)}}{2\cdot \:5}[/tex]

Now,

[tex]\sqrt{\left(-4\right)^2-4\cdot \:5\left(-1\right)} = \sqrt{\left(-4\right)^2+4\cdot \:5\cdot \:1}[/tex]

= [tex]\sqrt{16+20} = \sqrt{36} = 6[/tex]

Therefore
[tex]x_{1,\:2}=\frac{-\left(-4\right)\pm \:6}{2\cdot \:5}[/tex]

[tex]x_1=\frac{-\left(-4\right)+6}{2\cdot \:5}[/tex] = [tex]\frac{-\left(-4\right)+6}{2\cdot \:5} =[/tex]  [tex]\frac{10}{2\cdot \:5}[/tex] [tex]= 1[/tex]

[tex]x_2 = \frac{-\left(-4\right)-6}{2\cdot \:5}[/tex] [tex]= \frac{4-6}{2\cdot \:5}[/tex] = [tex]-\frac{1}{5}[/tex]

To find corresponding values for [tex]y_1 y_2[/tex] plug these values of [tex]x_1, x_2[/tex] into the line equation

[tex]y_1 = 1- 2(1) = -1[/tex]
[tex]y_2 = 1 - 2(-\frac{1}{5}) = 1 + \frac{2}{5} = \frac{7}{5}[/tex]

So the two points of intersection are

[tex](1, -1)[/tex] and ([tex](-\frac{1}{5}, \frac{7}{5})[/tex].

In decimal form this would be (1, -1) and (-0.2, 1.4)

See attached graph

Answer:

Student B

Step 1: the quantity 1 over sine theta end quantity minus sine theta over the quantity cosecant theta

Step 2: the quantity 1 minus sine squared theta end quantity over sine theta all over cosecant theta

Step 3: cosine squared theta over sine squared theta

Step 4: cot2 θ

Step-by-step explanation:

Answers:

4 liters of the 18% acid

8 liters of the 45% acid

========================================================

Explanation:

Let's call the bottles A and B

x = amount, in liters, of bottle A used

12-x = amount, in liters, of bottle B used

Bottle A has 0.18x liters of pure acid

Bottle B has 0.45(12-x) liters of pure acid

These amounts must add to 0.36*12 = 4.32 liters of pure acid since we want 12 liters of a 36% solution.

So,

0.18x + 0.45(12-x) = 4.32

0.18x + 0.45(12)+0.45(-x) = 4.32

0.18x + 5.4-0.45x = 4.32

-0.27x + 5.4 = 4.32

-0.27x = 4.32 - 5.4

-0.27x = -1.08

x = (-1.08)/(-0.27)

x = 4 liters which is the amount of bottle A needed

12-x = 12-4 = 8 liters is the amount of bottle B needed

--------------------

Check:

Using 4 liters of bottle A means we contribute 4*0.18 = 0.72 liters of pure acid from this bottle alone.

Using 8 liters from bottle B adds another 8*0.45 = 3.6 liters of pure acid

Then, 0.72+3.6 = 4.32 which matches with the 4.32 mentioned earlier. Therefore, the answer is fully confirmed.

The equation of the bisector of angle OAB is y=x

If E is the point of intersection of this bisector and line through A and B then the co-ordinates of E is (40/3,40/3)

Since E is the point of intersection of the bisector of angle OAB and line A and B  hence we can say OA:OB=AE:EB

Since point O is the origin hence we can say the co-ordinates of point O are (0,0) and the angle bisector of the angle OAB makes and angle of 45 degree from x axis, So by the formula of Line passing through origin (y= mx )

where ,

∅=angle made by the bisector the angle OAB to the x axis(i.e. 45 degree)

m=tan∅

now using substitution method ,

m=tan45

m=1

the equation of the bisector of angle OAB is  y=x

Now, the equation of the line passing through point A(5,0) and point B(0,8) by formula,

(y-y1)={(y2-y1)/(x2-x1)}*(x-x1)

here , y1=0, x1=5, y2=8, x2=0

again by substitution method,

(y-0)={(8-0)/(0-5)}*(x-5)

y={8/-5}*(x-5)

8x - 5y = 40

point E is the intersection of 8x - 5y = 40 and y=x the solving these two equations by substitution method we get;

x=40/3

y=40/3

the co-ordinates of point E are  (40/3,40/3)

E is the intersection point of the bisector of the angle OAB and the line through point A and B ,

on comparing triangle OEA and triangle OEB we can say that,

OA:OB= AE:EB

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The measure of the sides is 27, 23 and 16 and the perimeter is 56 respectively

The triangle is a 2 dimensional shape that has 2 sides.

Given the following sides

l = m + n - 12

n = m - 7

m = l - 1/4n

l = m + 1/4n

Substitute

m + 1/4n = m + m - 7 - 12

m + 1/4(m-7) = 2m - 19

m + 1/4m - 7/4 = 2m - 19

-m + 1/4m = -19 + 7/4

-3/4m = -19(4)+7/4

-3/4m = -69/4

-3m = -69

m = 23

Hence the value of m from the given expression is 23

n = 23 - 7

n = 16

l = 23 + 16 -12

l = 39 - 12

l = 27

n = 23 - 7

n = 16

Hence the measure of the sides is 27, 23 and 16 respectively

Perimeter = 27+23+16

Perimeter = 56

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Answer:

done it step by step

Step-by-step explanation:

with the original length as 15.17 feet and the new length as 18.57 feet, there is an increase of 22.41 % in the length of the garage.

We are given that Yossi has a garage of area = 230 feet²

We know that area of square is:

A = side²

230 = s²

s = √ 230

s = 15.17 feet

Now area is increased by 50 %

New area = 230 + 50% of 230

A' = 230 + 115

A' = 345 feet²

s² = 345

s = √ 345

s = 18.57 feet

Percent increase in length = 18.57 - 15.17 / 15.17 × 100

Percent increase in length = 22.41 %

Therefore, with the original length as 15.17 feet and the new length as 18.57 feet, there is an increase of 22.41 % in the length of the garage.

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The reciprocal of the divisor 5 1/5 divided by 1/10 is 10.

The reciprocal of the divisor: Reciprocal and division of fractions are two different methods. When the numerator and denominator of a fraction are interchanged, then it is said to be its reciprocal. Suppose a fraction is a/b, then its reciprocal will be b/a. A fraction is a numerical quantity that is not a whole number.

The reciprocal of the divisor 5 1/5 divided by 1/10

To the reciprocal of the divisor.

Equation: 5 1/5 divided by 1/10

1st: Flip the 1/10 to become 10/1

2nd: Change division to multiplication

New equation: 5.1/5.10/1

⇒ 50/5

       ⇒ 10.

The reciprocal of the divisor 5 1/5 divided by 1/10 is 10.

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Antonio can make 6 whole loaves of bread, and he will be left with 1/2 cup of flour, which is used to make 3/4th of a loaf.

In the question, we are given that Antonio is making bread. He has 9/2 cups of flour. We are also informed that each loaf uses 2/3 cup of flour.

We are asked for the number of whole loaves of bread that Antonio can make, and also, we are asked whether he will be left with some flour or not. If yes, then how much?

To find the number of loaves Antonio will make, we will divide the quantity he has by the quantity required for one loaf.

Thus, the number of loaves = (9/2) ÷ (2/3) = 9/2 × 3/2 = 27/4 = 6 3/4.

We will ignore the fractional part as we are asked for whole loaves. Thus, we can say that Antonio will make 6 whole loaves.

Also, he will be left with 3/4th of a loaf, that is, 3/4 of 2/3 cup of flour = 1/2 cup of four.

Thus, Antonio can make 6 whole loaves, and he will be left with 1/2 cup of flour, which is used to make 3/4th of a loaf.

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Eli's strawberry milk has a stronger strawberry taste than Arjun.

According to the question we have been given that

Arjun ,       adds milk = 90ml

                in strawberry syrup = 20ml

Eli,             adds milk = 180ml

                in strawberry syrup = 45ml

We need to find whose strawberry milk has stronger strawberry taste that is we have to find the concentration of the solution.

The formula for the concentration is given as :

   Concentration of the solution = [tex]\frac{volume of solute}{volume of solution}[/tex]  * 100%

For Arjun,  

        Volume of solvent = 90ml

      Volume of solute = 20ml

     Volume of solution = 20+90 ml = 110 ml

Concentration = (20/110)*100%

                        = 18.18%

For Eli,

         Volume of solvent = 180ml

      Volume of solute = 45ml

     Volume of solution = 180+45 ml = 225 ml

Concentration = (45/225)*100%

                        = 20%

As we can see that the Eli's concentration of the solution is greater than Arjun's. Hence Eli's strawberry milk has a stronger strawberry taste.

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Because the angles are vertical angles, the value of x is 1

The angles are given as

A = 7x + 26

B = 4x + 29

The angles are vertical angles

So, we have

A = B

This gives

7x + 26 = 4x + 29

Evaluate the like terms

3x = 3

Divide by 3

x = 1

Hence, the value of x is 1

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The number of children wearing both black and orange costumes is 7.

Let P denote the universal set, the number of children going to a party.

Then n(P) = 46

Let B denote the number of children wearing black costumes to the party.

Then n(B) = 18

Let O denote the number of children wearing orange costumes to the party.

Then n(O) = 12

We are asked to find how many children will wear both orange and black. That is to find n(B ∩ O).

We are also given that half of the guests are wearing costumes that are neither black nor orange.

Then n(B ∪ O)' = n(P)/2 = 46/2 = 23, where n(B U O)' denotes the complement of n(B U O).

We also know that n(B ∪ O) = n(P) - n(B U O)' = 46 - 23 = 23

⇒ n(B U O) = n(B) + n(O) - n(B ∩ O)

⇒ 23 = 18 + 12 - n(B ∩ O)

⇒ 23 = 30 - n(B ∩ O)

⇒ n(B ∩ O) = 30 - 23 = 7

⇒ n(B ∩ O) = 7.

Therefore, the number of children wearing both black and orange costumes is 7.

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The correct options are (a) , (14,3) , (d), (-8,6) and (e) , (-6,2)

By concept of translation, the coordinates of new triangle A'B'C' are given as follows:

A' (14, 3), B' (-6,2), C (-8, 6).

A translation can be define or represented by a change in the graph of a function, which are done by the means of operations such as multiplication, addition or subtraction either in it’s range(i.e. the values of y) or in it’s domain(i.e. the values of x). Examples are compression of points or shifting  left right or bottom up, stretching and reflections over the axis that is the x-axis or the y-axis, or rotations around the origin.

For the given question, the translation rule is given as follows:

(x , y) -> (2x, y-3).

Applying the rule to each point of triangle, we have that:

A': (2(7), 6-3) = (14, 3).

B': (2(-3), 5-3) = (-6, 2).

C': (2(-4), 9-3) = (-8, 6).

Thus, the coordinates of triangle formed i.e. A'B'C' are given as follows:

A' (14, 3), B' (-6,2), C' (-8, 6).

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The complete question is:

Triangle ABC lies on the coordinate plane with vertices located at A(7,6), B(-3,5), and C(-4,9). The triangle is transformed using the rule (x,y) -> (2x,y-3) to create triangle A'B'C'. Select all possible answers for the vertices of triangle A'B'C'.

Question 1 options:

a. (14,3)

b. (9,3)

c. (-6,10)

d. (-8,6)

e. (-6,2)

the other statements are.

For a conditional statement of the form:

If P, then Q

The other statements are:

Here the conditional is:

"if a figure is a rectangle with sides 2 cm and 3cm, then it has a perimeter of 10cm"

We can identify:

P = a figure is a rectangle with sides 2 cm and 3cm

Q = it has a perimeter of 10cm

Then the other statements are.

This clearly is false, as we can have a circle with a perimeter of 10 cm.

This is also false, the figure can be a rectangle with sides of 1cm and 4 cm for example.

This is true. if the figure does not have a perimeter of 10cm, then it can't be a rectangle with sides of 2cm and 3cm

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The sum of the expression in simplest form is 8( x + 2y)

Algebraic expressions are expressions made up of variables, terms, factors, constants and coefficients.

They are also made up of mathematical operations such as addition, subtraction, multiplication, division, etc

Given the expressions;

(3x+9y) and (5x+7y)

Let's find the sum

3x + 9y + 5x + 7y

collect like terms

3x + 5x + 9y + 7y

8x + 16y

Find common factor

8( x + 2y)

Thus, the sum of the expression in simplest form is 8( x + 2y)

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The volume of a rectangular pyramid is 1020 units is 5.67 units

The volume of a rectangular pyramid is 1020 units.

volume of rectangular pyramid = LBH

L= length of the rectangular base

B=  width of the rectangular base

H=  height of the pyramid

here L= 10units B= 18 units

V= LBH

H=V/(LB)

  = 1020/(10)(18)

  = 5.67 units

Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units or by various imperial units. The definition of length is interrelated with volume

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Answer: -1.4,1 1/25,1.25

Step-by-step explanation:

-1.4,1 1/25,1.25

1 1/25=

1+0.04=1.04

Order: -1.4, 1.04, 1.25 ==> -1.4,1 1/25,1.25

Juan is paid 1575 pesos, 1800 pesos, 2137.5 pesos

It is given that Juan Dela Cruz is paid 45 peso for every hour he works for up to 40 hours and 1.5 times i.e. 67.5 pesos for every hour after 40 hours.

The first part asks for Juan's earning if he works 35 hours

Therefore , according to the function,

Juan's pay = 35 x 45 = 1575 pesos

The second part asks for Juan's earning for 40 hours

Therefore, according to the function,

Juan's pay for 40 hours = 40 x 45 =1800 pesos

The third part asks for Juan's earning for 45 hours

Juan's pay for 45 hours according to the function=

67.5h-900 = 67.5 x 45 -900 = 2137.5 pesos

Hence, Juan is paid 1575 pesos, 1800 pesos, 2137.5 pesos

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Statement (C) "Lynn's method will not give her the correct answer. Multiplying by 4 and then doubling the product is the same as multiplying by 8, not by 44" is the correct explanation of the above-given situation.

So, 4 × 305 = 1220

Therefore, statement (C) "Lynn's method will not yield the correct result. Multiplying by 4 and then doubling the result is equivalent to multiplying by 8, not 44 "is the correct explanation for the situation described above.

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The complete question is given below:

Lynn wants to multiply 44 × 305 44×305. She says that all she has to do is find 4 × 305 4×305 and then double that number. Which explains whether or not Lynn's method will give her the correct answer?

A. Lynn's method will give her the correct answer. Multiplying by 4 and then doubling the product is the same as multiplying by 44.

B. Lynn's method will not give her the correct answer. Multiplying by 4 and then doubling the product is the same as multiplying by 6, not by 44.

C. Lynn's method will not give her the correct answer. Multiplying by 4 and then doubling the product is the same as multiplying by 8, not by 44.

D. Lynn's method will not give her the correct answer. Multiplying by 4 and then doubling the product is the same as multiplying by 16, not by 44.

Since the rate of change is constant, the relation is linear, and is modeled by:

y = 4x - 4.

A linear function is modeled by:

y = mx + b

In which:

For this problem, we have that when x changes by 1, y always changes by 4, hence we have a linear relation with a slope of 4, given as follows:

y = 4x + b

When x = 1, y = 0, hence we use it to find b as follows:

0 = 4 + b

b = -4.

Hence the relation is:

y = 4x - 4.

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Answer: 1/2 * n

1 hour of practice in 2 days

2 hours of practice in 4 days

5 hours of practice in 10 days

Step-by-step explanation:

60 minutes=1 hour

30 minutes * 1 hour/60 minutes=

30*1/60=

30/60=1/2 hour.

1/2 * n

1/2 * 2 = 1 hour of practice in 2 days

1/2 * 4 =

4/2 = 2 hours of practice in 4 days

1/2 * 10=

10/2 = 5 hours of practice in 10 days

Using a system of equations, it is found that it sold 27 lower-priced speaker docks.

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

For this problem, the variables are given as follows:

Last week the store sold three times as many lower-priced speaker docks as higher-priced speaker docks, hence:

x = 3y -> y = x/3.

Combined sales totaled $3,870, hence, considering the price of each speaker:

80x + 190y = 3870.

Since x = 3y:

240y + 190y = 3870.

430y = 3870

y = 3870/430

y = 9.

Hence:

x = 3y = 3 x 9 = 27.

It sold 27 lower-priced speaker docks.

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Answer:

Step-by-step explanation:

10 would be the total, since 2/10 are green. 1/2 of 10 is 5, so 5 candles are pink. 5+2=7. 10-7=red candles (3), there is ultimately 3 candles that are red.

The direction of the resultant vector is 58 degrees east-north if the mouse walks 15 m east, makes a 90° left turn, walks 8 m, makes another 90° left turn, and walks 10 m.

It is defined as the quantity that has magnitude as well as direction also the vector always follows the sum triangle law.

It is given that:

A mouse is trapped in a maze.

He walks 15 m east, makes a 90° left turn, walks 8 m, makes another 90° left turn, and walks 10 m.

Based on the data given:

tanA = 8/(-10+15) = 8/5

A = tan⁻(8/5)

A = 58 degrees

The direction will be 58 degrees east-north

Thus, the direction of the resultant vector is 58 degrees east-north if the mouse walks 15 m east, makes a 90° left turn, walks 8 m, makes another 90° left turn, and walks 10 m.

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Based on the numbers given, we can say that 51 is a multiple of 3.

51 is not a prime number which means that it has factors and it is a multiple of several numbers.

The factors of 51 include:

1, 3, 17, and 51

Out of the numbers given of 2,3,5,6,9,10, it therefore means that 51 is a multiple of only one number which is 3. When 3 is multiplied by 17, it gives 51.

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The rule that best describes the transformation is (x, y) = (x - 3, y)

The points of the triangle have the coordinates

(2, 4), (5, 3), and (7, 2).

The image of the triangle have the coordinates

(−1, 4), (2, 3), and (4, 2)

When the above points and the image are compared, we can see that the y-coordinates remain unchanged

This means that the triangle is translated horizontally

Using points (2, 4) and (-1, 4);

We have

(x, y) = (-1 - 2, 4 - 4)

Evaluate

(x, y) = (-3, 0)

Rewrite as

(x, y) = (x - 3, y)

Hence, the rule that best describes the transformation is (x, y) = (x - 3, y)

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The contribution of the first data point, (54, 60) to correlation is given by the option with the equation equation;

The correlation coefficient is given by the formula;

[tex]r = \frac{1}{n - 1} \times \sum( \frac{x - \bar{x}}{s_{x} })( \frac{y - \bar{y}}{s_{y} })[/tex]

Where;

(x, y) are the coordinates of a data point.

[tex] \bar{x}[/tex] = The mean for the x-values

[tex]S_{x}[/tex] = The standard deviation of the x-values

[tex] \bar{y}[/tex] = The mean of the y-values

[tex]S_{y}[/tex] = The standard deviation of the y-values

The given parameters are;

[tex] \bar{x}[/tex] = 58.4

[tex]S_{x}[/tex] = 16.87

[tex] \bar{y}[/tex] = 60.8

[tex]S_{y}[/tex] = 21.89

The formula for the contribution of the first data point, (54, 60) to correlation, r, is therefore;

[tex] \displaystyle{\left( \frac{54 - 58.4 }{16.87}\right)\cdot \left( \frac{60 - 60.8}{21.89 }\right)}[/tex]

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Answer:

Yes, he will have more than enough flags, 1/3*3 flags is 1 mile, the race is only 9/10 miles.

Step-by-step explanation: