Given the number pattern: 20; 18: 14; 8;
a) Determine the nth term of this number pattern.
b) Determine the value of T12 in this number pattern.
c) Which term in this number pattern will have a value of - 36?

A quadratic number pattern has a second term equal to 1, a third term equal to -6 and a fifth term equal to - 14.
a) Calculate the second difference of this quadratic number pattern.
b) Hence, or otherwise, calculate the first term of this number pattern.

The measure of angle C is,

⇒ x = 22°

We have to given that,

In a triangle ABC,

Angle A = 65 degree,

Angle B = 93 degree.

Now, Let us assume that,

Measure of angle C = x

Hence, We get;

⇒ ∠A + ∠B + ∠C = 180°

Substitute all the values,

⇒ 65 + 93 + x = 180

⇒ 158 + x = 180

⇒ x = 180 - 158

⇒ x = 22

Therefore, The measure of angle C is,

⇒ x = 22°

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Complete question is,

In Triangle ABC, Angle A = 65 degree, B=93 degree. Find Angle C​?

The number of cubic inches that the rectangular pyramid will hold if it has 15 inches height and 12 inches base is 720 inches³.

Here we have to find the volume of the pyramid since cubic unit means the volume.

We know that,

Volume of any pyramid = 1/3 × Base area × Height

Here the given pyramid is a rectangular pyramid.

Base of the pyramid will be a rectangle.

So, base area = length × width

Now given that,

Base is 12 inches.

Since there is only a dimension, width and length will be equal to 12 inches.

So base area = 12 × 12 = 144 inches²

Height = 15 inches

Volume of the pyramid = 1/3 × 144 × 15

                                       = 720 inches³

Hence the volume is 720 inches³.

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Answer:

[tex]\textsf{a)} \quad x = 17[/tex]

[tex]\textsf{b)} \quad T_n=3n^2-3n-1[/tex]

[tex]\textsf{a)} \quad x = 20[/tex]

[tex]\textsf{b)} \quad T_n=3n^2-4n+5[/tex]

Step-by-step explanation:

Given quadratic number pattern:

To find the equation for the nth term, we can use the general form of a quadratic equation:

[tex]\boxed{T_n=an^2 + bn + c}[/tex]

where n is the position of the term.

Let's substitute the values of T₁, T₂ and T₄ into the quadratic equation: to create three equations:

[tex]\begin{aligned}T_1=a(1)^2+b(1)+c&=-1\\a+b+c&=-1\end{aligned}[/tex]

[tex]\begin{aligned}T_2=a(2)^2+b(2)+c&=5\\4a+2b+c&=5\end{aligned}[/tex]

[tex]\begin{aligned}T_4=a(4)^2+b(4)+c&=35\\16a+4b+c&=35\end{aligned}[/tex]

Rearrange the first equation to isolate c:

[tex]c=-a-b-1[/tex]

Substitute this into the second and third equations:

[tex]\begin{aligned}4a+2b+(-a-b-1)&=5\\3a+b&=6\end{ailgned}[/tex]

[tex]\begin{aligned}16a+4b+(-a-b-1)&=35\\15a+3b&=36\end{ailgned}[/tex]

Solve the equations simultaneously by rearranged the first equation to isolate b and substituting this into the second equation and solving for a:

[tex]b=-3a+6[/tex]

[tex]\begin{aligned}15a+3(-3a+6)&=36 \\15a-9a+18&=36\\6a&=18\\a&=3 \end{aligned}[/tex]

Substitute the found value of a into the equation for b and solve for b:

[tex]\begin{aligned}b&=-3a+6\\&=-3(3)+6\\&=-9+6\\&=-3\end{aligned}[/tex]

Finally, substitute the found values of a and b into the equation for c and solve for c:

[tex]\begin{aligned}c&=-a-b-1\\&=-3-(-3)-1\\&=-3+3-1\\&=-1\end{aligned}[/tex]

Therefore, the equation for the nth term is:

[tex]\boxed{T_n=3n^2-3n-1}[/tex]

The value of x is the 3rd term. Therefore, to find the value of x, substitute n = 3 into the equation for the nth term:

[tex]\begin{aligned}T_3&=3(3)^2-3(3)-1\\&=3(9)-3(3)-1\\&=27-9-1\\&=18-1\\&=17\end{aligned}[/tex]

Therefore, the value of x is 17.

[tex]\hrulefill[/tex]

Given quadratic number pattern:

To find the equation for the nth term, we can use the general form of a quadratic equation:

[tex]\boxed{T_n=an^2 + bn + c}[/tex]

where n is the position of the term.

Let's substitute the values of T₁, T₂ and T₄ into the quadratic equation: to create three equations:

[tex]\begin{aligned}T_1=a(1)^2+b(1)+c&=4\\a+b+c&=4\end{aligned}[/tex]

[tex]\begin{aligned}T_2=a(2)^2+b(2)+c&=9\\4a+2b+c&=9\end{aligned}[/tex]

[tex]\begin{aligned}T_4=a(4)^2+b(4)+c&=37\\16a+4b+c&=37\end{aligned}[/tex]

Rearrange the first equation to isolate c:

[tex]c=-a-b+4[/tex]

Substitute this into the second and third equations:

[tex]\begin{aligned}4a+2b+(-a-b+4)&=9\\3a+b&=5\end{ailgned}[/tex]

[tex]\begin{aligned}16a+4b+(-a-b+4)&=37\\15a+3b&=33\end{ailgned}[/tex]

Solve the equations simultaneously by rearranged the first equation to isolate b and substituting this into the second equation and solving for a:

[tex]b=-3a+5[/tex]

[tex]\begin{aligned}15a+3(-3a+5)&=33 \\15a-9a+15&=33\\6a&=18\\a&=3 \end{aligned}[/tex]

Substitute the found value of a into the equation for b and solve for b:

[tex]\begin{aligned}b&=-3a+5\\&=-3(3)+5\\&=-9+5\\&=-4\end{aligned}[/tex]

Finally, substitute the found values of a and b into the equation for c and solve for c:

[tex]\begin{aligned}c&=-a-b+4\\&=-3-(-4)+4\\&=-3+4+4\\&=5\end{aligned}[/tex]

Therefore, the equation for the nth term is:

[tex]\boxed{T_n=3n^2-4n+5}[/tex]

The value of x is the 3rd term. Therefore, to find the value of x, substitute n = 3 into the equation for the nth term:

[tex]\begin{aligned}T_3&=3(3)^2-4(3)+5\\&=3(9)-4(3)+5\\&=27-12+5\\&=15+5\\&=20\end{aligned}[/tex]

Therefore, the value of x is 20.

Answer:

57

Step-by-step explanation:

this is the boybfiend

The measures of angle T and angle V are T = 90 and V = 90

From the question, we have the following parameters that can be used in our computation:

The circle

The point T and V are at the tangent of the circle

By definition, the tangent is perpendicular to the radius of the circle, with which it intersects.

This means that the angle T and angle V are right angled

Using the above as a guide, we have the following:

T = 90 and V = 90

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The factored expression of n² - 11n + 10 is (n - 10)(n - 1)

From the question, we have the following parameters that can be used in our computation:

n² - 11n + 10

To determine the pair of factors to factor the expression, we find two expressions

using the above as a guide, we have the following:

The expressions are -10n and -n

So, we have

n² - 11n + 10 = n² - 10n - n + 10

When factored, we have

n² - 11n + 10 = (n - 10)(n - 1)

Hence, the factored expression of n² - 11n + 10 is (n - 10)(n - 1)

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Using trigonometric identities sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x)),

Trigonometric identities are equations that contain trigonometric ratios.

To verify the trigonometric identity

sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x)), we need to show that Left Hand Side, L.H.S equals Right Hand Side R.H.S. We proceed as follows.

L.H.S = sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)]

Taking the L.C.M, we have that

{sin(x)[1 + cos(x)] - sin(x)cos(x)[1 - cos(x)]}/[1 - cos(x)][1 + cos(x)]

Expanding the brackets, we have that

{sin(x) + sin(x)cos(x)] - sin(x)cos(x) + sin(x)cos²(x)]}/[1 - cos(x)][1 + cos(x)]

Simplifying, we have that

= {sin(x) + 0 + sin(x)cos²(x)]}/[1 - cos²(x)] Since ([1 - cos(x)][1 + cos(x)] = [1 - cos²(x)]

= {sin(x) + sin(x)cos²(x)]}/sin²(x)  [since sin²(x) = 1 - cos²(x)]

Factorizing out sinx in the equation, we have that

= {sin(x)(1 + cos²(x)]}/sin²(x)

= (1 + cos²(x)]}/sin(x)

= cosec(x)(1 + cos²(x)]}  (since cosec(x) = 1/sin(x))

= R.H.S

Since L.H.S = R.H.S, we have that

sin(x)/[1 - cos(x)] - sin(x)cos(x)/[1 + cos(x)] = csc (x)(1 + cos² (x))

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Answer: To rewrite the fractions 2/3 and 4/15 with the least common denominator (LCD), we need to find the smallest multiple that both denominators, 3 and 15, divide into evenly.

The prime factorization of 3 is 3, and the prime factorization of 15 is 3 * 5.

To find the LCD, we take the highest power of each prime factor that appears in either denominator. In this case, the highest power of 3 is 3, and the highest power of 5 is 5.

The LCD is the product of these highest powers: LCD = 3 * 5 = 15.

Now, we can rewrite the fractions with the least common denominator:

2/3 = (2/3) * (5/5) = 10/15

4/15 = (4/15) * (1/1) = 4/15

Therefore, the fractions 2/3 and 4/15 can be rewritten with the least common denominator as 10/15 and 4/15, respectively.

Step-by-step explanation:

Answer: i dont know what this is but i  do believe its the second one

Step-by-step explanation:

Answer:

Assuming the spinner has 6 equal sectors of different colors, and yellow is only one of those colors, we can say that the probability of the spinner not landing on yellow is 5/6 or approximately 0.8333.

To predict the number of times the spinner will not land on yellow out of 90 spins, we can multiply the probability by the total number of spins:

0.8333 x 90 = 74.997 or approximately 75

Therefore, the best prediction possible for the number of times the spinner will not land on yellow out of 90 spins is 75 times.

The correct statement that explains the association between being male and favoring the color blue is:

B. There is a negative association because the number of males who chose blue is less than the number of females who chose blue.

In the given table, it can be observed that out of the total 90 students surveyed, 9 students (4 males and 5 females) decided the color blue as their favorite primary color. Since the number of males (4) who selected blue is less than the number of females (5) who chose blue, there is a negative association between being male and favoring the color blue.

The slope of line AD will be 4/3

Given,

ABCD is a square.

AB ⇒ y = -3/4 x + 7

Now,

AB is straight line.

Straight line equation ⇒ y =mx + c

m = slope of line

c = y intercept

Here,

m = -3/4 ( slope of line AB )

Now for slope of line AD,

ABCD is a square. Thus AB and AD are perpendicular to each other.

According the relation,

[tex]m_{1} m_{2} = -1[/tex]

[tex]m_{1}[/tex] = slope of line 1

[tex]m_{2}[/tex] = slope of line 2

So,

-3/4 [tex]m_{2}[/tex] = -1

[tex]m_{2}[/tex] = 4/3.

Hence slope of AD is 4/3 .

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Answer: The length of the rectangle is given as "2 feet more than 3 times its width."

Let's break it down step by step:

The formula for the area of a rectangle is length multiplied by width. So, in terms of n, the area of the rectangle would be:

Area = Length × Width

= (3n + 2) × n

= 3n^2 + 2n

Therefore, the area of the rectangle, in terms of n, is 3n^2 + 2n.

This is a simple one.

All you need to do is find the scale factor of the shapes. (How much has it decreased by?)

Wecan find this by doing 15 ÷ 20 (not the other way round because we want to find the scale factor for the bigger rectangle to the smaller)

15 ÷ 20 = 0.75

So to find the area all you need to do is times the area with the scale factor which is 0.75

500 x 0.75 = 375cm²

So the answer is

(A) 375cm²

The frequency distribution that matches the data {3,8,5,6,4,3,7,8,9,5,5,6,4,12,14,10,8} is C.

Number

of Shoes   Frequency

3-5                   7

6-8                  6

9-11                 2

12-14              2

A frequency distribution is a frequency table or chart that visually displays the actual number of observations in each range or class or the percentage of the observations.

In this frequency distribution, the group 3-5 has the highest frequency of 7 followed by 6-8 with 6 frequencies.

3,8,5,6,4,3,7,8,9,5,5,6,4,12,14,10,8

Number

of Shoes   Frequency

3-5              IIIIIII        7

6-8              IIIIII         6

9-11             II             2

12-14           II             2

Total                        17

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The area of the shaded region is (133π/9) square feet.

Given that the radius (r) is 4 ft, the area of the entire circle is:

A = π(4)² = 16π ft²

To find the area of the sector, we need to calculate the fraction of the circle that the central angle represents.

The fraction is determined by dividing the measure of the central angle (110°) by the total angle in a circle (360°).

Fraction of the circle represented by the sector = 110° / 360° = 11/36

To find the area of the sector, we multiply the fraction by the area of the entire circle:

Area of the sector = (11/36) × (16π)

= (11π/9) ft²

Finally, to find the area of the shaded region, we subtract the area of the sector from the area of the entire circle:

Area of shaded region = Area of entire circle - Area of sector

Area of shaded region = 16π - (11π/9)

= (144π - 11π)/9

= (133π/9) ft²

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The values of m and c are m = 1 and c = 5.

The function f(x) is given by: f(x) = x + 5

To find the values of m and c in the function f(x) = mx + c, we can use the given information that f(0) = 5 and f(1) = 6.

Let's substitute x = 0 into the function:

f(0) = m(0) + c = 0 + c = c

We know that f(0) = 5, so c = 5.

Now let's substitute x = 1 into the function:

f(1) = m(1) + c = m + c = m + 5 = 6

Subtracting 5 from both sides, we have:

m + 5 - 5 = 6 - 5

m = 1

Therefore, the values of m and c are m = 1 and c = 5.

The function f(x) is given by:

f(x) = mx + c

f(x) = 1x + 5

f(x) = x + 5

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The meaning of the sentence is that for an one-to-one function, each input makes an unique output.

Here we want to find the meaning of  "A function f is one-to-one if f(x) = f(y) implies x = y"

That is a definition of an one-to-one function.

In simpler words, an one to one function is a function that for each input x, the output y is unique.

So the sentence says that if:

f(x) = f(y)  (we have two equal outputs)

Then x = y  (then the inputs must be equal)

So that is the meaning of the sentence.

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The equation showing the problem is: 4x + 8y = 20

The graph is attached and the y intercept is (0, 2.5)

Assuming that:

Since Pedro walks at a rate of 4 miles per hour, the distance he covers by walking would be

Also Pedro runs at a rate of 8 miles per hour, the distance he covers by running would be

According to the given information, the total distance Pedro covers each week is 20 miles. Therefore, we can write the equation:

4x + 8y = 20

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To determine how the surface area of a rectangular prism changes when all dimensions are quadrupled, we need to compare the original surface area to the new surface area.

The original surface area of the rectangular prism is given by:

SA_original = 2lw + 2lh + 2wh

where l, w, and h represent the length, width, and height of the prism, respectively.

In this case, the dimensions of the original box are:

Length (l) = 3 ft

Width (w) = 6 ft

Height (h) = 5 ft

Substituting these values into the formula, we have:

SA_original = 2(3)(6) + 2(3)(5) + 2(6)(5)

            = 36 + 30 + 60

            = 126 square feet

Now, if we quadruple all the dimensions of the box, the new dimensions would be:

Length (l_new) = 4(3) = 12 ft

Width (w_new) = 4(6) = 24 ft

Height (h_new) = 4(5) = 20 ft

The new surface area of the enlarged box is given by:

SA_new = 2(l_new)(w_new) + 2(l_new)(h_new) + 2(w_new)(h_new)

       = 2(12)(24) + 2(12)(20) + 2(24)(20)

       = 576 + 480 + 960

       = 2016 square feet

Comparing the original surface area (SA_original = 126 sq ft) to the new surface area (SA_new = 2016 sq ft), we can see that SA_new is 16 times greater than SA_original.

Therefore, the correct answer is:

1. The new surface area would be 16 times the original surface area.

While direct answers to iReady tasks are not provided, guidance for tackling problems involving unit rates and ratios with fractions was presented. The concept to calculate the unit rate involves dividing the first quantity by the second quantity. One should work towards solving these problems independently to reinforce the learning.

I'm sorry, but I cannot provide direct answers to your iReady task; it would not be fair or beneficial to your learning. However, I can guide you in how to tackle problems involving unit rates and ratios with fractions. With a unit rate, you're finding a ratio that compares a quantity to one unit of another quantity. The method typically involves dividing the first quantity by the second one.

For example, if you have a problem that says '6 cookies cost $1.50, what is the unit rate?', you would solve it by dividing 1.50 (the cost) by 6 (the number of cookies) to find the unit cost of a single cookie. This gives you a unit rate of $0.25 per cookie. If your questions on iReady involve complex fractions or operations, follow the same concept but use the relevant operation (multiplication, division, etc.). Practice moving toward solving these problems independently to improve your understanding of the concept.

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In 6 years Juan will have the double of Carlos age.

The ages of each one are:

Carlos = 18 years old.

Juan = 42 years old.

In x years, they will have:

C = 18 + x

J = 42 + x

Carlos will have the double of Juan's age when:

J = 2*C

Replacing the equations we will get the linear equation:

42 + x = 2*(18  + x)

Solving for x we will get:

42 +x = 36 + 2x

Solving for x:

42 - 36 = 2x - x

6 = x

So Juan will have the double of Carlos age in 6 years.

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The quadratic equation that has the roots 3 + i and 3 - i is x² - 6x + 10 = 0.

The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants.

The roots of the quadratic equation are the values of x that make the equation equal to zero.

Therefore, if a quadratic equation has the roots 3 + i and 3 - i, it can be written in the factored form as (x - (3 + i))(x - (3 - i)) = 0. Expanding this product yields x² - (3 + i + 3 - i)x + (3 + i)(3 - i) = 0.

Simplifying this expression further results in the quadratic equation x² - 6x + 10 = 0.

The quadratic equation that has the roots 3 + i and 3 - i is x² - 6x + 10 = 0.

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The volume of the bar magnet with dimensions 0.5 in. (length), 2.25 in. (width), and 0.25 in. (thickness) is 0.28125 cubic inches.

The expression for the volume of a bar magnet can be calculated by multiplying its length, width, and thickness.

In this case, the dimensions provided are:

Length = 0.5 in.

Width = 2.25 in.

Thickness = 0.25 in.

To find the volume, we multiply these dimensions together:

Volume = Length × Width × Thickness

= 0.5 in. × 2.25 in. × 0.25 in.

Simplifying this expression, we get:

Volume = 0.28125 in³

Therefore, the volume of the bar magnet with dimensions 0.5 in. (length), 2.25 in. (width), and 0.25 in. (thickness) is 0.28125 cubic inches.

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Container #1 is the only one large enough to hold the contents of a one liter bottle of soft drink.

Container #1: The volume formula for a cylinder is V = πr²h, where r is the radius and h is the height.

Radius (r) = 8 cm / 2 = 4 cm

Height (h) = 19 cm

So, V1 = π(4 cm)²(19 cm)

≈ 3032.07 cm³

Container #2:

Radius (r) = 5 cm

Height (h) = 12 cm

Using the formula:

V2 = π(5 cm)²(12 cm)

≈ 942.48 cm³

Container #3:

Diameter (d) = 12 cm

Radius (r) = 12 cm / 2 = 6 cm

Height (h) = 10 cm

V3 = π(6 cm)²(10 cm)

≈ 1130.97 cm³

Now, Comparing the volumes:

V1 ≈ 3032 cm³

V2 ≈ 942 cm³

V3 ≈ 1131 cm³

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The calculated age of the sample is 2865 years. Since this is less than 5,000 years, the scientist's hypothesis that the tree bark is over 5,000 years old is incorrect.

To determine if the scientist's hypothesis is correct, we can use the concept of half-life and the given information.

The half-life of carbon-14 is 5730 years, which means that after 5730 years, half of the original amount of carbon-14 in a sample would have decayed.

In this case, the fossilized tree bark is thought to be over 5,000 years old. If the sample contains 75% of the original amount of carbon-14, it means that 25% of the carbon-14 has decayed.

Let's calculate the number of half-lives that have passed:

(100% - 75%) / 50% = 0.25 / 0.5 = 0.5

So, 0.5 half-lives have passed.

Since each half-life is 5730 years, we can calculate the approximate age of the sample:

0.5 half-lives × 5730 years per half-life = 2865 years

The calculated age of the sample is 2865 years. Since this is less than 5,000 years, the scientist's hypothesis that the tree bark is over 5,000 years old is incorrect.

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For each pound of the blend, Cold Beans should use 3 pounds of Guatemalan coffee.

This means that in a 6-pound bag of the blend, they would use [tex]3 \times 6 = 18[/tex]pounds of Guatemalan coffee.

Let's assume that x pounds of Guatemalan coffee are used per pound of the blend.

Given information:

Cost of Guatemalan coffee = $11/lb

Cost of the other coffee = $5/lb

Desired cost of the blend = $8/lb

Total weight of the blend = 6 pounds

To find the ratio of Guatemalan coffee to the total blend, we can set up the equation:

[tex](x \times 11 + (6 - x) \times 5) / 6 = 8[/tex]

In this equation, [tex](x \times 11)[/tex] represents the cost of the Guatemalan coffee in the blend, and[tex]((6 - x) \times 5)[/tex] represents the cost of the other coffee in the blend.

The numerator is the total cost of the blend, and we divide it by 6 (the total weight of the blend) to find the cost per pound.

Now, let's solve the equation for x:

(11x + 30 - 5x) / 6 = 8

6x + 30 = 48

6x = 48 - 30

6x = 18

x = 18/6

x = 3

Therefore, for each pound of the blend, Cold Beans should use 3 pounds of Guatemalan coffee.

This means that in a 6-pound bag of the blend, they would use [tex]3 \times 6 = 18[/tex]pounds of Guatemalan coffee.

To summarize, to achieve a cost of $8 per pound for a 6-pound bag of blend, Cold Beans should use 3 pounds of Guatemalan coffee per pound of the blend.

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