For which triangle does the Pythagorean Theorem
express the relationship between the lengths of its three
sides?
A
B
C
D

The parent function for the quadratic equation is y = x²

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

The graph

From the graph, we have the following points

vertex = (0, 0)

Points = (±1, 1) and (±2, 4)

The equation for the quadratic function is represented as

y = a(x - h)² + k

Where vertex = (h, k)

So, we have

y = a(x - 0)² + 0

Evaluate

y = ax²

Using the points, we have

a(1)² = 1

Evaluate

a = 1

So, we have

y = x²

Hence, the parent function for the quadratic equation is y = x²

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The area of the sector is approximately 9.25 ft².

We have,

The formula for the area of a sector.

A = (θ/360°) × πr²

where A is the area of the sector, θ is the central angle in degrees, and r is the radius.

Substituting the given values.

A = (40°/360°) × 3.14 × (16.7 ft)²

A = 0.1111 × 3.14 × 278.89 ft²

A = 9.2465 ft²

Rounding to the nearest hundredth.

A ≈ 9.25 ft²

Therefore,

The area of the sector is approximately 9.25 ft².

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The trigonometric value equation is solved and sin ( 0 )° + cos ( 0 )° = 1

Given data ,

Let the trigonometric relation be represented as A

Now , the value of A is

sin ( 0 )° - cos ( 0 )° = 1

From the trigonometric relation , we get

sin 0 = 0

cos 0 = 1

So , the equation is

sin ( 0 )° + cos ( 0 )° = 1

Hence , the equation is solved

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Answer:

Part A: The point z1 is at (4, 6) and the point z2 is at (-5, 2). The distance between them is √((4-(-5))^2 + (6-2)^2) = √(9^2 + 4^2) = √(81 + 16) = √97.

Part B: The complex conjugate of z2 is (-5, -2). Geometrically, it can be found by reflecting the point z2 across the x-axis.

Part C: To find z2 − z1 geometrically, you can draw a vector from z1 to z2. The coordinates of this vector are (-5-4, 2-6) = (-9, -4). So z2 − z1 = -9 - 4i.

The given system of equations has no solution.

Given that a system of equations, 12x + 36y = −3 and −12x − 36y = 0, we need to solve it,

We know for a system of equations =

a₁x + b₁y = c₁ and a₂x + b₂y = c₂

If a₁ / a₂ = b₁ / b₂ = c₁ / c₂ then the system has infinite solutions.

If a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂ then the system has no solutions.

If a₁ / a₂ ≠ b₁ / b₂ then the system has a unique solution.

comparing the coefficients given,

-12/12 = -1

-36/36 = -1

-3/0 = -3

We get here the condition of a₁ / a₂ ≠ b₁ / b₂, which means that the system has no solution.

Hence the given system of equations has no solution.

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Answer:Therefore, the dimensions of the rectangle are 14 ft by 4 ft.

Let's use variables to represent the length and width of the rectangle. We know from the problem that:Length = 3 times the width + 2

Area = 40 ft²We can use the formula for the area of a rectangle to set up an equation:Area = length x widthSubstituting in the values we have:40 = (3w + 2)wExpanding the brackets and simplifying:40 = 3w² + 2wRearranging into a quadratic equation form:3w² + 2w - 40 = 0We can solve for w using the quadratic formula:w = (-2 ± sqrt(2² - 4(3)(-40))) / (2 x 3)

w = (-2 ± sqrt(484)) / 6

w = (-2 ± 22) / 6The two possible solutions are w = 4 and w = -6.67. Since width cannot be negative, we take the positive solution: w = 4 ft.To find the length, we can use the formula we established earlier:Length = 3w + 2

Length = 3(4) + 2

Length = 14 ft

1. The product of MG is as follows;

[tex]\left[\begin{array}{ccc}1&k_1+9&4k_2+36 \\2t_1&k_1t_1-17&3k_2-19t_1-11\\t_2+1&k_1-4t_2+5&k_2t_2-8\end{array}\right][/tex]

2. The values are k₁ = -9, k₂ = -9, t₁ = 0 and t₂ = -1 given that G = M⁻¹

Matrix M and matrix G are given as

M = [1, 4, 5; t₁, 3, -1; 1, t₂, 1] and

G = [2, k₁, -19; 1, -4, k₂; -1, 5, 11]

MG becomes

1×2 + 4×1 + 5×-1         1k₁+4×-4+ 5×5       1×-19+4k₂+5×11

t₁×2+3×1+-1×-1             t₁k₁+3×-4+-1×5       t₁×-19+3k₂+-1×11

1×2+t₂×1+1×-1               1k₁+t₂×-4+-1×5        1×-19+t₂k₂+1×11

=

1             k₁+9         4k₂+36

2t₁          k₁t₁-17        3k₂-19t₁-11

t₂+1        k₁-4t₂+5      k₂t₂-8

To solve G = M⁻¹ we know it is an identity matrix

I = | 1 0 0 |

    | 0 1 0 |

    | 0 0 1 |

We can equate the elements of MG and I to find the values of t₁, t₂, k₁, and k₂:

1 = 1,         k₁+9 = 0,       4k₂+36 = 0

2t₁ = 0     k₁t₁-17 = 1        3k₂-19t₁-11 = 0

t₂+1 = 0     k₁-4t₂+5 = 0   k₂t₂-8 = 1

k₁+9 = 0           4k₂+36 = 0            2t₁ = 0          t₂+1 = 0

k₁ = -9                 k₂ = -9                   t₁ = 0           t₂ = -1

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The statement thet complete the proof of the congruent triangles is (a) SAS criterion

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

The incomplete proof

In the proof, we have

AD ≅ BC --- side length

DC ≅ DC --- side length

∠ADC ≅ ∠BCD -- angle

This means that the triangles are congruent by the SAS theorem

The SAS theorem is defined by "If two sides in one triangle are congruent to two sides in another triangle and the included angle in both are congruent, then the two triangles are congruent"

This means that they are congruent by the SAS congruent

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Answer: A. SAS criterion.

Step-by-step explanation: Angle : ∠ADC ≅ ∠BCD                                         Side length : AD ≅ BC                                                                                       Side length : DC ≅ DC                                                                                                

Because of this △ADC ≅ △BCD by SAS criterion.

Answer:  [tex]g^2-9g+14[/tex]

Step-by-step explanation:

(g-7)(g-2)

[tex]g^2 - 2g-7g+14[/tex]

[tex]g^2-9g+14[/tex]

Use FOIL method.

The first derivative of the parametric curve is:

dx/dt = [tex]-1/(t-1)^2[/tex] and dy/dt = [tex](2t(t-1) - t^2)/(t-1)^2.[/tex]

1. To find the first derivative of the parametric curve x = 1/(t-1) and [tex]y = t^2/(t-1)[/tex], we can use the chain rule.

2. Let's differentiate each component separately.

3. Differentiating x with respect to t, we have:

dx/dt = [tex]-1/(t-1)^2[/tex]

4. Differentiating y with respect to t, we have:

dy/dt = [tex](2t(t-1) - t^2)/(t-1)^2[/tex]

5. Therefore, the first derivative of the parametric curve is:

dx/dt = [tex]-1/(t-1)^2[/tex] and dy/dt = [tex](2t(t-1) - t^2)/(t-1)^2.[/tex]

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Answer:

  80°

Step-by-step explanation:

You want the sum of angles 1 and 2 in the attached figure where line m ║ line n.

Vertical angles are congruent. This means angle 1 is 60°, and the unnumbered angle in triangle ABC at vertex A is 20°.

Where a transversal crosses parallel lines, alternate interior angles are congruent. This means angle 2 is congruent with the unnumbered interior angle at A, which we know is 20°.

Angle 1 is 60° and angle 2 is 20°. Their sum is ...

  ∠1 +∠2 = 60° +20° = 80°

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The error E = ± 4.04 %

Sample data = ( p ).

success p = success percentage = 40 %

confidence interval CI = 98%

sample size n = 800

margin of error E:

The margin of error "E" for estimation of population proportion ( p ) is given by:

E = z - critical √(p(1-p)/n)

P ( Z < Z-critical ) = a/

a = 1 - CI

P ( Z < Z-critical ) = (1 - 0.98) / 2

P ( Z < Z-critical ) = 0.01

Z-critical = 2.33

The error E = ± 4.04 %

Thus, the correct answer is E=± 4.04 %

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The angles are formed by a transversal and two parallel lines in the figure. By applying congruent angles and vertical opposite angle property, ∠1 ≅∠3 ≅ ∠5 ≅ ∠7; ∠2 ≅ ∠4 ≅ ∠6 ≅ ∠8. Therefore option B is correct.

Congruent angles are identical to each other and have the same measure. The angles formed by the transversal in corresponding corners are called corresponding angles and they are congruent.

Therefore, ∠1 ≅ ∠5, ∠3 ≅ ∠7----------(1)

At a vertex if two angles are opposite to each other, then they are vertically opposite angles, they are equal and congruent to each other.

Therefore, ∠1 ≅ ∠3, ∠5 ≅ ∠7----------(2)

Combining (1) and (2), we get:

∠1 ≅∠3 ≅ ∠5 ≅ ∠7
Similarly we can find,

∠2 ≅ ∠4 ≅ ∠6 ≅ ∠8.

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The total number of berry pitchers made is 15/5 or 3.

If the serving size of berry pitchers is 1/5, it means that each pitcher represents 1/5 of a full serving. To determine the total number of pitchers made, we need to calculate how many times 1/5 fits into the whole serving.

Let's assume that the total number of pitchers made is represented by "x". We can set up the following equation:

x * 1/5 = 3

To solve for "x," we multiply both sides of the equation by 5:

5 * (x * 1/5) = 5 * 3

This simplifies to:

x = 15/5

Thus, the total number of berry pitchers made is 15/5 or 3.

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[tex]w^2 - 6w + 4[/tex] is a polynomial that represents how much more profit is earned by selling w apple trees than w pear trees.

The profit earned by selling w pear trees is 2w + 10

To find the difference in profit, we need to subtract the profit earned by selling w pear trees from the profit earned by selling w apple trees:

[tex](w^2 - 4w + 14) - (2w + 10)[/tex]

Simplifying the expression:

[tex]w^2 - 6w + 4[/tex]

Therefore, the polynomial that represents how much more profit is earned by selling w apple trees than w pear trees is [tex]w^2 - 6w + 4[/tex].

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The solution to the equation  [tex]2log_a(x) - log_a(x-1) = log_a(x-2)[/tex]  is

x = 2/3.

To solve the equation [tex]2log_a(x) - log_a(x-1) = log_a(x-2)[/tex] for x, We can employ algebraic manipulation and logarithmic characteristics.

Apply logarithmic properties.

Using the logarithmic property[tex]log_a(b) - log_a(c) = log_a(b/c),[/tex] we can rewrite the equation as:

[tex]log_a(x^2) - log_a(x-1) = log_a(x-2).[/tex]

Combine the logarithms.

Applying the logarithmic property [tex]log_a(b) - log_a(c) = log_a(b/c),[/tex] we can combine the logarithms on the left-hand side:

[tex]log_a[(x^2)/(x-1)] = log_a(x-2).[/tex]

Take the logarithms out.

We can do away with the logarithms because the base of the equation is the same.

[tex](x^2)/(x-1) = x-2.[/tex]

Streamline the equation.

We can eliminate the denominator and simplify the equation by multiplying both sides by (x-1):

[tex]x^2 = (x-2)(x-1).[/tex]

Expand and simplify.

Expanding the right-hand side, we have:

[tex]x^2 = x^2 - 3x + 2[/tex]

Rearrange the equation.

Rearranging the equation, we get:

0 = -3x + 2.

Do the x math.

We must isolate x on one side of the equation in order to find its value. Taking 2 away from both sides:

3x = 2.

Finally, divide both sides by 3:

x = 2/3.

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From matrix property it can be showed that A=B.

Given,

If A,B and x are matrices with x≠0 and AX=BX

Now,

X must be non-singular matrix because

A[tex]XX^{-1}[/tex]=B[tex]XX^{-1}[/tex]

AI = BI

= I( identity matrix )

⇒A = B

Hence Proved, A = B .

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The expression that represents the surface area of the cube is 25 + 25 + 25 + 25 + 25 + 25

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

The cube

The side length of the cube is

Side length = 5

The surface area of the cube is calculated as

SA = 6 * Side length²

So, we have

SA = 6 * 5²

Expand

SA = 6 * 25

Further expand

SA = 25 + 25 + 25 + 25 + 25 + 25

Hence, the expression that represents the surface area of the cube is 25 + 25 + 25 + 25 + 25 + 25

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Complete question

The following figure shows a cube. Which expression represents the surface area of the cube? Choose 1 answer

(a) 5 5 - 5 - 5

(b) 25 + 25 + 25 + 25 + 25 + 25

(c) 5 + 5 + 5 + 5 + 5 + 5

The figure is attached

I agree that Nicole made a mistake because the total cost of the Five year plan is $ 34, 000 and not $ 74, 000.

The total cost of the Five year plan would be:

= Payment per month for the five year plan x Number of months in year x number of years + Down payment

= 400 x 12 x 5 + 10, 000

= 4, 800 x 5 + 10, 000

= $ 34, 000

The mistake that Nicole made was to multiply the $ 14, 800 that was the cost of the five year plan in one year, by five years. This was wrong because it meant including a new down payment in every one of the five years instead of just once.

She should have multiplied $4, 800 by 5 years and then added the down payment which is paid just once.

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The x-value on the line when y = 100 represents the estimated number of hours a student would need to study to achieve a quiz grade of 100.

We have,

Assuming that the line of best fit represents the trend between the number of hours students study and their quiz grades,

The x-value on the line when y = 100 represents the estimated number of hours a student would need to study to achieve a quiz grade of 100.

In other words,

If a student studies for this amount of time, they are predicted to achieve a quiz score of 100 based on the trend shown by the line of best fit.

Thus,

The x-value on the line when y = 100 represents the estimated number of hours a student would need to study to achieve a quiz grade of 100.

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The equation of h(x) in vertex form is given as follows:

h(x) = (x + 2)² - 3.

The quadratic function of vertex(h,k) is given by the rule presented as follows:

y = a(x - h)² + k

In which:

The vertex is the turning point of the function, where it changes from decreasing to increasing, or vice versa, hence it's coordinates are given as follows:

(-2,-3).

Then the equation is:

h(x) = a(x + 2)² - 3.

When x = 0, h(x) = 1, hence the leading coefficient a is obtained as follows:

1 = a(2)² - 3

4a = 4

a = 1.

Hence the equation is:

h(x) = (x + 2)² - 3.

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The equation of the quadratic function C(t) is given as follows:

C(t) = t² + 3.

The quadratic function of vertex(h,k) is given by the rule presented as follows:

y = a(x - h)² + k

In which:

The vertex is the turning point of the function, where it changes from decreasing to increasing, or vice versa, hence:

(0,3).

Then:

y = at² + 3.

When x = 1, y = 4, hence the leading coefficient a is obtained as follows:

4 = a(1)² + 3

a = 1.

Hence the function is given as follows:

y = t² + 3.

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The correct statements are:

If "positive" end behaviour:

The arrows point in the same direction.

The right arrow is up.

The graph hits the y-axis.

If "negative" end behaviour:

The arrows point in opposite directions.

The right arrow is down.

The graph hits the X-axis.

In mathematics, a graph is a visual representation of a set of data, which consists of points, called vertices or nodes, and lines connecting them, called edges or arcs.

Graphs are often used to represent mathematical functions, data sets, or relationships between objects, and they can be used to model real-world phenomena.

Graphs can be drawn on a coordinate plane or in a more abstract form, and they can be used to analyze and understand complex systems and relationships.

The correct statements are:

If "positive" end behaviour:

The arrows point in the same direction.

The right arrow is up.

The graph hits the y-axis.

If "negative" end behaviour:

The arrows point in opposite directions.

The right arrow is down.

The graph hits the X-axis.

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The horizontal transformations are defined as follows:

cos(ax).

They are classified as follows:

The vertical transformations are defined as follows:

acos(x).

They are classified as follows:

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To achieve a 1 in row 1, column 1, divide row 1 by 4 in the given matrix, resulting in the updated matrix. This row operation preserves the system of linear equations represented by the matrix.

To get a 1 in row 1, column 1 of the given matrix, you can perform the following valid row operation:

R1 = (1/4) * R1

This operation scales row 1 by a factor of 1/4, resulting in the new matrix:

1 , -3/4 , -43/4

3 , -2 , -30

By dividing each element in row 1 by 4, the value in row 1, column 1 becomes 1. This row operation maintains the equivalence of the system of linear equations represented by the matrix.

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The exponential equation that models the estimated population of sage grouse in Silverton County is: [tex]y = 115,000 * (1/4)^x[/tex]

Given that the population is expected to decrease by a factor of 1/4 each decade, we know that the common ratio, b, is 1/4.

Since the most recent estimate is about 115,000 sage grouse, we can assign this value to the initial population, a.

Therefore, the exponential equation that models the estimated population of sage grouse in Silverton County is: [tex]y = 115,000 * (1/4)^x[/tex]

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a. The standard deviation of the tuition is  $150.

b. The shape of this distribution is a normal distribution, which is a bell-shaped curve.

(a) The mean tuition is $100 per credit hour * 15 credit hours per semester + $150 lab fee = $1650 per semester.

The standard deviation of the tuition is l:

= ✓(100² + 150²) =

= ✓22500

= $150.

(b) The shape of this distribution is a normal distribution, which is a bell-shaped curve. The mean is the center of the distribution, and the standard deviation is the width of the distribution. The majority of the tuitions will be close to the mean, with fewer and fewer tuitions as you move further away from the mean.

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The amount Sam needs to pay in each installment is $141.1

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

Cost of refrigerator = 1058

Down payment = 211.40

Number of equal installments = 6

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

Each installment = (Cost of refrigerator - Down payment)/Number of equal installments

substitute the known values in the above equation, so, we have the following representation

Each installment = (1058 - 211.40)/6

Evaluate

Each installment = 141.1

Hence, the amount in each installment is $141.1

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